newton's method and periodic solutions of nonlinear wave equations

Newton’s Method and Periodic Solutions of Nonlinear Wave Equations

WALTER CRAIG Brown University

AND

C . EUGENE WAYNE Pennsylvania State Universi5

Abstract

We prove the existence of periodic solutions of the nonlinear wave equation

a:u = a.:u - g k u ) ,

satisfying either Dirichlet or periodic boundary conditions on the interval 10, rr]. The coefficients of the eigenfunction expansion of this equation satisfy a nonlinear functional equation. Using a version of Newton’s method, we show that this equation has solutions provided the nonlinearity g(x, u ) satisfies certain generic conditions of nonresonance and genuine nonlinearity. 01993 John Wiley & Sons. Inc.

1. Introduction

In this paper we address the existence problem for time periodic solutions of nonlinear wave equations of the form

(1.1) ah = 8:u - g(x, u ) .

The solutions u(x, t ) are defined on the spatial interval 0 5 x 5 T , satisfying either periodic (u(x + T , ? ) = u(x,t)), or Dirichlet (u(0,t) = 0 = u(7r,t)) boundary conditions. This is called the problem of free vibrations of the nonlinear string, and has received much attention over the last twenty-five years; an extensive set of references appears in the paper of H. Brezis; see [3]. In this paper we prove the existence of families of periodic solutions to ( 1 . 1 ) in a neighborhood of the equilibrium solution u = 0, provided that the nonlinearity g ( x , u ) is analytic and satisfies certain conditions of nonresonance and genuine nonlinearity. These conditions have their classical analog in the theory of dynamical systems, and we show that they hold for a set of nonlinearities that is open and dense in the natural topology. The difficulty in the construction is that one must overcome the phenomenon of small denominators that is present in this problem. Our approach is through a novel method based on the Nash-Moser technique and a Lyapunov- Schmidt decomposition, which is described below.

The first mathematical results concerning (1.1) were for the problem of forced vibrations, which appeared in an article by P. Rabinowitz; see [ 191. In this paper,

Communications on Pure and Applied Mathematics, Vol. XLVI. 1409-1498 (1993) 0 1993 John Wiley & Sons. Inc. CCC 0010-3640/93/111409-90

1410 W. CRAIG AND C. E. WAYNE

solutions are constructed for a wave equation similar to ( l . l ) , which has explicit periodic dependence on time, where the ratio of temporal period to spatial period is restricted to be rational. The latter restriction is equivalent to a compactness condition, and plays an important role in the proof. At about the same time there appeared a paper of J. B. Keller and L. Ting (see [ l l ] ) , on free vibrations for the nonlinear Klein-Gordon equation, which while non-rigorous has a certain resemblance to our present approach to the problem. It was realized at the time (by, e.g., J. J. Stoker, K. 0. Friedrichs, and J. Moser) that the study of free vibrations for solutions in a neighborhood of an equilibrium exhibited the phenomenon of small denominators; see [ 151. The first mathematical advances for free vibrations were by P. Rabinowitz in [20], in which he posed the problem globally as an indefinite variational principle. Under the same assumption of rationality of the ratio of space and temporal periods, and under certain conditions on the nonlinear term g(x, u), he proved the existence of solutions to (1.1) as critical points. Many authors used similar global variational techniques to obtain related results; for example, H. Brezis, J.-M. Coron, and L. Nirenberg in [4]. All of these results use strongly the compactness that comes from the rationality condition on the periods of the solutions.

More recently, S . Kuksin in [12] and [13] and E. Wayne in [22] have given a perturbative approach to the problem which is based on an infinite dimensional extension of the theorem of Kolmogorov, Arnold, and Moser (KAM). This provides constructions of solutions to the free vibrations problem locally in a neighborhood of u = 0, which can be both periodic and quasiperiodic. These solutions typically have their temporal and spatial periods rationally independent. Additionally, B. V. Lidskij and E. I. Shulman in [ 141 have found a family of time periodic solutions for (1.1) in the case that g = u3; the frequencies taken on by this family form a Cantor set of zero Lebesgue measure.

In this paper we adopt a different approach to the local perturbation problem, which is related to the Lyapunov-Schmidt method of bifurcation theory. The infinite dimensional component of the problem is solved by a variant of the Nash- Moser method, as it must address issues raised by the presence of small denominators. Our theorem applies to an open and dense set of nonlinearities g(x, u) - in fact the condition of genuine nonlinearity only fails on a set which is essentially of codimension 1. For a given admissible nonlinearity, Cantor families of periodic solutions are obtained, corresponding to sets of frequencies which also form Cantor sets of positive measure. One advantage of this approach as opposed to more traditional KAM methods is that there is no reliance on an initial partial Birkhoff normal form, so that we can handle certain resonant cases; see [7]. We think that this novel approach to small denominator problems is robust, and is an interesting alternative to the classical approach to the KAM theorem. There is a restriction on the nonlinearity in our results that it be an analytic function of both arguments. This is not required in [12], and we believe that it is not an essential requirement for our techniques. It does, however, considerably simplify the proof.

Equation ( 1 . 1 ) describes a dynamical system, and can be written as a Hamil- tonian system in infinitely many degrees of freedom. The usual Hamiltonian

PERIODIC SOLUTIONS OF NONLINEAR WAVE EQUATIONS 141 1

function is the energy

where d,,G = g. To briefly describe our approach in this paper, denote the Taylor expansion of the nonlinear term by g(x, u) = gl(x)u + g2(x)u2 + g3(x)u3 + . . .. The equation (l.l), when linearized about the zero solution u = 0, has solutions obtained simply by separation of variables, in terms of eigenfunction-eigenvalue pairs {+,(x), us} of the Sturm-Liouville operator L(g l ) = (- $ + gl (x)), with the appropriate boundary conditions. These eigenfunctions are a convenient basis for the configuration space of the nonlinear dynamical system (1.1). To construct time periodic solutions with frequency 0, substitute < = Or and expand the function u(x ,E) in this basis and its Fourier series in <,

u(x, <) = C i(j, k)+,(x)eik< . j.l.

With this definition of u(x, r ) , (1.1) is equivalent to a nonlinear lattice problem for the coefficients { h ( j , k ) } . If we assume that L J ~ > 0, then a family of periodic solutions of the linear problem, with frequency w,/k, is given by

where b ( j , k) is the Kronecker &function at the lattice site ( j , k) E Z+ x Z. These are approximate solutions to the nonlinear problem, and we prove that an iteration scheme based on this first approximation converges to a solution of the nonlinear equation (1.1). The linearized lattice problem, however, typically has spectra that are dense on the real axis, a reflection of the small denominators in the problem. Due to this fact, the iteration scheme converges for only a Cantor subset of parameters I-, albeit one of positive measure. The result is the construction of a set of periodic orbits whose frequencies fill a Cantor set of positive measure near the linear frequency w,. The parametrization of the family of nonlinear solutions by the linear solutions is analogous to the Lyapunov-Schmidt method. Here the infinite dimensional part of the problem is solved using the technique of Nash-Moser. An interesting feature is that the method of inversion of linear operators that is central to this technique is closely related to the methods used to study spectral problems for lattice Schrodinger operators, developed by J. Frohlich and T. Spencer in their work on localization theory; see [9]. From a technical viewpoint, this connection between localization theory and dynamical systems strikes us as one of the most interesting aspects of our approach. A similar connection, with applications to differential-difference equations, has been previously discussed in the work of C. Albanese, J. Frohlich, and T. Spencer; see [ I ] and [2].

The existence theorem does not apply to all choices of nonlinearity g. It requires that certain conditions of linear nonresonance and genuine nonlinearity are


satisfied. Analogs of these conditions occur classically in the theory of dynamical systems. Our conditions depend only upon the 3-jet of g, and are finite in number. The set of nonlinearities which satisfy them is generic, indeed it is open and dense, and is described more precisely in Section 7. These conditions can in principle be checked in explicit cases. For the classical examples of the nonlinear Klein-Gordon and the sine-Gordon equations, the nonlinearity depends upon one parameter, and we show that, for an open set of full measure of this parameter, the conditions are satisfied and the existence theorem applies.

Although we focus on the nonlinear wave equation in this paper, it is expected that similar constructions exist for the nonlinear Schrodinger equation, the generalized KdV equation, and other conservative evolution equations for which the construction of periodic solutions encounters the small denominator problem.

Our results for the nonlinear wave equation (1.1) give an infinite dimensional version of the Lyapunov center theorem, whose classical setting is for finite dimensional Hamiltonian systems in a neighborhood of an elliptic equilibrium point. Suppose for a finite dimensional problem that the frequencies of the system, linearized about equilibrium, are Wj; j = 1, . . . , N . If the nonresonance condition

holds for all integer pairs ( j , k ) # (1, l), then kR - Wj # 0 for an interval of frequencies R around W l , and the Lyapunov theorem asserts that a smooth family of periodic solutions bifurcates from the family of solutions of the linearized system, with frequency R close to the linearized frequency Wl. In contrast, if we focus on a frequency we, for the infinite dimensional problem, (1.3) does not preclude that the quantites kwe, - w, are dense, a reflection of the fact that there is a small denominator problem even for periodic solutions. Our results are that under a sufficient replacement of (1.3), there also exist families of periodic solutions of the nonlinear problem close to the linear solutions. That is, we construct a smooth curve bifurcating from the family of solutions of the linear equation, in a neighborhood of equilibrium. We cannot, however, prove that all points on the curve give rise to solutions of the nonlinear equation, but only that a Cantor set of parameters of positive measure corresponds to solutions. As the frequency parameter R varies, there is in general a dense set of frequencies in resonance, that is kR - w, = 0 for some integers ( j , k ) . It is the process of excising these and analogous resonances which gives rise to the Cantor set of frequencies for which our iteration scheme converges.

There is an additional comparison of our results with the theorem of A. We- instein (see [23]) in the resonant case, where there are multiple solutions of

kwe, - wj = 0 ; ( j , k ) E 2’ .

The above method is sufficiently robust to handle these cases as well, as long as there is only finite multiplicity. The result is an existence theorem for Cantor families of periodic solutions, with frequencies near the linear frequency we,.

PERIODIC SOLUTIONS OF NONLINEAR WAVE EQUATIONS 1413

Furthermore, we are able to estimate the number of such families. This is the subject of another paper (see [7]) to which we refer for details.

As remarked above, the KAM approach to this problem also yields the existence of quasi-periodic solutions for equations like (1.1). We believe that an extension of the present method will also yield quasi-periodic solutions, and plan further work on this problem. Such results would be of interest even for finite dimensional systems, since they would give a proof of the existence, near an elliptic equilibrium point, of invariant tori for Hamiltonian systems whose dimension is less than or equal to the number of degrees of freedom of the system; this proof would differ from that of 181 and [16].

We conclude with an outline of the remainder of the paper. In the next section we state our principal results, transform the wave equation to a problem on a two- dimensional lattice, and apply our results to discuss two well-known examples, the Klein-Gordon and sine-Gordon equations. In Section 3 we state the induction hypotheses which allow us to derive the results in Section 2. Section 4 contains the verification of these induction hypotheses, while Section 5 explains how to control the inverse of the linearized operator which arises in Newton’s method. This analysis is connected with the theory of localization in Schrodinger operators. Section 6 is devoted to verifying that all relationships between the parameters of the induction which were used to control the convergence of the iteration scheme can be simultaneously satisfied. Finally, in Section 7 we derive some estimates that are used in linking the wave equation ( 1 . 1 ) to the lattice problem. In this final section we also discuss the issues of genericity of the nonlinearity.

2. Results

2.1. Nonlinear Wave Equations

The equation we study in this paper is the nonlinear wave equation on a bounded interval 0 5 x S 7r in one space dimension

(2.1) ah4 = a.:u - g(x,u) .

We assume that the nonlinear term is analytic in both variables in the region { ( x , u ) I IIm x( < 5, IuI < I } , periodic in x with period T , and with Taylor expansion in u,

(2 .2)

We are seeking solutions which are periodic in time, with period 27r/R, which satisfy certain self-adjoint boundary conditions on a spatial interval. Linearizing about the solution u = 0 we obtain

(2 .3)

g ( x , u ) = gl (x)u + g2(x)u2 + g3(x)u3 + ’ . ‘ .

a:v = a.:v - g l ( x ) v .

Two examples of boundary conditions that we will address are Dirichlet conditions, u(0, t ) = 0 = U ( T , t ) , and periodic conditions u(x + 7r, 1) = u(x, t ) . Solutions


of the linear equation (2.3) are given by separation of variables and an eigenfunction expansion. Let { + b , ( ~ ) } j ” = ~ be normalized eigenfunctions for the the linear differential operator

(2.4)

with the proper boundary conditions, with eigenvalues <w;>,”= (For periodic boundary conditions, it is more convenient to begin labeling the eigenvalues and eigenfunctions with j = 0.) Solutions of (2.3) are given by

which are parametrized by the amplitudes rj and the phases [ j . For real wj each function cos(w,t + J)$j(x) is time periodic, with frequency w,. A more general solution to the linear equation (2.3) is time periodic only if for all nonzero amplitudes rj there exists a full set of rational relations between the associated frequencies wj; that is, there exists w and integers k j such that for all j with r j # 0, w, = k,w. Unless a full set of resonance conditions are satisfied the general solution is quasiperiodic or almost periodic. We fix an index j , and seek periodic solutions to the full nonlinear problem (2.1) near the family of linear solutions

with frequency near the real linear frequency wj. In this process any coincidence or near coincidence of linear frequencies causes resonance and other phenomena related to small divisors in the full nonlinear problem. The following results, however, demonstrate that for most nonlinearities g(x, u) an iterative construction can overcome these difficulties, to prove the existence of periodic solutions to (2.1) of small amplitude, which are perturbations of the linear solutions (2.6).

THEOREM 2.1. Consider equation (2.1) with periodic boundary conditions on the interval 0 5 x S 7 ~ . For an open dense set of nonlinear terms g(x,u) there exist time periodic solutions close to (2.6). More precisely i f g ( x , u) is in this set, there exists r* > 0, a Cantor set $? C LO, r* ) of positive measure, and a C” function a(r), with R(0) = wj, such that for every r E %‘, there exists a periodic solution u(x, t; r ) of (2.1) with frequency Wr) . These solutions are analytic in x and t and satisfy

(2.7)


We note that not only will the set of amplitudes r of the solutions we construct have positive measure. but the set of frequencies of the periodic solutions will also have positive measure, thus assuring the existence of solutions of irrational period.

To be precise about the topology on the nonlinear terms that appear in Theo- rems 2.1 and 2.2, let d~ be the set of bounded analytic functions on {(x , u) I IIm X I < a. I u I < 1 }, periodic in x with period 7r, and satisfying g(x, 0) = 0. The topology used in this description is the Co topology on d ~ .

The conditions on the nonlinear term g(x, u ) in Theorem 2.1 and Theorem 2.2 are quite explicit, depending only upon the coefficients gl(x), g?(x), g3(x), in other words only upon the 3-jet of g. Roughly, there is a condition on gl in order to avoid certain primary resonances in the linear equation, and a condition of genuine nonlinearity placed upon gl, g2, and g3. Both are open conditions, excluding sets of finite codimension. Unfortunately the case gl(x) = 0 is too resonant for the present methods to handle, and is in the excluded set. On the other hand for an open set of constants mz of full Lebesgue measure the case gl(x) = m2 is included in the conditions of the theorems, thus the nonlinear Klein-Gordon equation and the sine-Gordon equations, and nonlinear perturbations of them, are covered by our results.

THEOREM 2.2. Consider equation (2.1) with Dirichlet boundar?, conditions. Additionally ask that g(x, u ) = -g(-x, -u). Among this class of nonlinearities there is an open dense set such that there exist time periodic solutions of (2.1) close to (2.6). More precisely i fg(x , u) is in this set, there exists r r > 0, a Cantor set g C [O,r,) of positive measure, and a C” function R(r), with Q(0) = w J , such that for every r E %, there exists a periodic solution U ( X , t ; r ) of (2.1) with frequency R(r). These solutions are analytic in x and t and satisb

lu(x. I ; r ) - r cos(Rt + <)+,(x)I < Cr’ (2.8)

J R - u , ~ < C r 2 .

The additional requirement of oddness of g ( . , . ) in Theorem 2.2 stems from the necessity to control the nonlinear coupling of eigenfunctions of (2.4). For Dirichlet boundary conditions, oddness of g reduces the estimates to the periodic case. We know of no general studies of the nonlinear interaction of eigenfunctions and we feel that this would be a worthwhile subject to pursue.

The inverse of the linearized operator 8; - 8: + gI(x) in (2 .3) plays a role in the existence results. When applied to time-periodic functions with frequency 0, the point spectrum of the operator is {w: - Q 2 k 2 1 1 5 j < x, -x < k < m}. For most choices of R and coefficient gl(x) this is a dense set in R. In particular, spectrum will accumulate at zero, a phenomenon which is called the small divisor problem. It is in this case that the results Theorem 2.1 and Theorem 2.2 are most interesting. The proof of these two results is by a Nash-Moser iteration scheme, giving a result which is in spirit very close to the theorem of Kolmogorov, Arnold, and Moser. In fact the conclusions are reminiscent of these results: we construct


families of solutions of (2.1) parametrized by r E C a Cantor set. These solutions are invariant circles in the phase space. That is, the periodic solutions that we find do not form smooth surfaces as they do in the finite dimensional Lyapunov center theorem, but rather they occur in totally disconnected families - Cantor sets of positive measure foliated by invariant circles. This gives a set of positive measure of amplitudes r for which there are solutions. This feature of totally disconnected families of solutions is familiar in finite dimensional problems, in the study of invariant tori of quasiperiodic orbits for Hamiltonian systems near elliptic stationary points. The Cantor structure of the families of periodic solutions that we construct is due to the fact that Hamiltonian systems possessing infinitely many degrees of freedom have the possibility for the generation of a dense set of linear resonances, even in the periodic case.

A straightforward consequence of the above results is the following infinite dimensional analogue of the Lyapunov center theorem.

COROLLARY 2.3. For either of the above boundary conditions there is a generic set Ce of nonlinearities such that for every g E Ce, every family of periodic solutions of the linear equation (2.3) gives rise to a (Cantor) family of periodic solutions to the nonlinear equation (2.1).

2.2. Nonlinear Lattice Problems

Both Theorem 2.1 and Theorem 2.2 follow from a more general result in the form of a Nash-Moser type theorem for nonlinear equations posed on the lattice Z+ x Z. Two-dimensional lattices are not special, and the theorem is easily generalized; in our situation two lattice directions conveniently index the temporal and spatial eigenfunctions used to describe the above problems in nonlinear waves.

The lattice problem arises by expressing the solutions u(x,t) of (2.1) in an expansion in eigenfunctions and Fourier series. The coefficients U(i, j ) in this expansion and the frequency parameter R E R must satisfy nonlinear equations on the lattice of the form

(2.9) W ( U ) + V(R)U = 0 .

Denoting lattice sites x = ( j , k ) E Z+ X Z and R E R a frequency parameter, the form of V ( R ) is a diagonal linear operator on sequences U(x),

(2.10)

where 6(x, y ) is the Kronecker delta. The sequence of frequencies {w,>j”=, satisfies

V(R)(x, y ) = (u; - R2k*)S(X, y ) .

which is the case for the eigenvalues of the linear operator (2.4) with either periodic or Dirichlet boundary conditions. We ask that the nonlinear term satisfies W(0) = 0, DuW(0) = 0, and furthermore that W ( U ) satisfy certain natural conditions of genuine nonlinearity, which we explain below.


Again we are led to linearize the nonlinear problem (2.9) about the solution U ( x ) = 0, obtaining

Solutions of this are simply p(x) = S,(x), with y = ( j , k ) , and R = ( u ~ , / k ) . Each nonzero eigenspace of V(R) is at least two dimensional, from the form of V(R) = (w: - R2k2), which is spanned by the vectors in e2(Z+ x Z) supported on the lattice sites y = ( j . k ) and y* = ( j , - k ) .

The nonlinear existence theorem focuses on solutions near the linearized solution space, with frequency near the value R = w, of the linearized problem. Without loss of generality we consider k = 1. Theorem 2.1 and Theorem 2.2 will hold for perturbations of periodic solutions of (2.3) associated with any positive eigenvalue. In fact with little loss of generality we assume that UJI is real and focus on a neighborhood of R = w1, with solutions supported near y = (1 , l ) and y* = (1, - 1). In general, operator (2.4) may have finitely many negative eigenvalues. The analysis in this paper is not affected by their presence.

In order to make the first step in the existence theorem we ask for certain conditions of nonresonance among the linear frequencies {w,>,X=, . This does not have to be a condition among infinitely many of them, but at least a large enough number of the initial frequencies. Let Lo be a positive integer and do a positive real number.

V(R)(P = 0 .

DEFINITION 2.4. A sequence {w,},”~ is (do, Lo)-nonresonant with w1 if there exists some T > 5 such that for all I j l + Ikl 5 Lo the following conditions hold:

(2.1 1)

and

(2.12)

PROPOSITION 2.5. Consider any L , > 0 and 1/2 < 77 < 1. Among the nonlinearities g ( . , . ) in (2.1) which have w: > 0, there is an open dense set such that for either periodic or Dirichlet boundary conditions the associated frequency sequence {w,} is (&‘,Lo) nonresonant with w1, for some LQ 2 L,.

In fact, the frequency sequence {wj} depends only on the 1-jet. that is, upon g$), not upon the whole nonlinear term. It is easy to see that for any j o , if w;” > 0, the proof of this proposition applies equally well to the case of linear problems with frequency w , ~ .

A major part of this paper is devoted to the analysis of the linearized operator


of (2.9). By an analogy with quantum mechanics we call H ( U ) a Hamiltonian operator, and the matrix of the inverse operator G(U)(z) = ( H ( U ) - z l ) - ' the Green's function. Central to the construction is the approximate inversion of H(U) about an approximate solution U . This involves an analysis of the small eigenvalues of H(U) , and the geometry of the lattice sites x at which V(R)(x,x) is close to zero. We define a singular site to be a lattice point x = ( j , k ) at which IV(O)(x,x)l = Iw! - 02k21 < d,. Any connected set of singular sites will be called a singular region. We show that by restricting the frequency R appropriately, singular regions for the Dirichlet problem consist only of isolated sites, while for the periodic problem singular regions will consist of no more than pairs of adjacent sites.

The constants d, 5 do, and Lo are all interrelated. We assume that IR2 - wit < do/2c . Then a frequency sequence which is (&,Lo) nonresonant gives an operator H ( U ) whose only singular sites in the lattice region Bo = { ( j , k ) E Z+ x Z I I j l + Ikl S Lo} are { ( j , k ) } = ((1, kl)}. Approximate solutions to the equation (2.9) are obtained by restricting the equations to the region Bo, giving a bifurcation problem that is solved by the Lyapunov-Schmidt method. An exact solution of (2.9) is obtained by an inductive procedure starting from this approximation. Convergence of the induction scheme depends on the correct choice of the constants r + , d,, do, and Lo. In particular we choose do = 2d, = for some

< 71 < 1, which, as we shall see below, results in the maximum amplitude of solutions r* LO(2+n)( l+F)

2.3. Symmetries of the Equation

The wave equation (2.1) has certain elementary properties of symmetry, relevant to this paper, that are reflected in the nonlinear lattice systems (2.9). The sequences U E e2(2+ X Z) among which we construct solutions are complex; they will, however, ultimately correspond to real solutions of the wave equation. Denote the involution on the lattice

(2.13) x = ( j , k ) - x* = ( j , - k )

and the complex conjugate of U by u. Then the reality condition on sequences is that U ( x ) = U ( x * ) . We shall require that the lattice equation (2.9) is covariant with respect to this symmetry,

__

(2.14)

so that when R E 58, the nonlinear operator of (2.9) preserves the class of sequences which satisfies the reality condition.

The wave equation respects an additional translational symmetry; t - t + T , T E R. That is, time translation leaves the equation and the boundary conditions invariant. We shall consider lattice systems which also possess a continuous


symmetry of this form. The translation operator on sequences in C2(Z+ X Z) is the diagonal operator

T { U ( X ) = eikEU(x) ,

where x = ( j , k ) . Tc is a unitary operator on t2(Z+ X Z). and it preserves the reality condition. From the nature of the diagonal operator notice that T:V(R)U = V(R)T,U. We shall further require that the nonlinearity W satisfy

(2.15) T,W(U) = W(T:U) .

The nonlinear terms in lattice problems arising from the nonlinear wave equation satisfy (2.15), because the system is autonomous. Our construction will be of families of solutions invariant with respect to this translation. The interpretation is that this is the construction of embedded invariant circles of solutions of (2.1) in the space t 2 ( Z + x 72).

The final requirement on the lattice problem is that when R is real and U satisfies the reality condition, the linearized operator H ( U ) = V(Q) + DW(U) is self-adjoint. Since V(R) will be real and diagonal, this is the condition on the nonlinear term W ( U ) that

Again this condition holds for problems stemming from the nonlinear wave equation. This assumption is tantamount to assuming that the original system is Hamil- tonian. It is only used, however, in certain estimates of the Green's function G(u)(z), and we believe that these can be replaced by alternate, if more difficult methods, and the condition of self-adjointness can be weakened.

2.4. Results for Lattice Problems

We work in a family of Hilbert spaces Zg C e2(Z+ x Z), which consist of elements of e2(Z+ x Z) for which the norm

is finite. If CJ = 0, then Rg = t2. We denote the inner product in f? by either (.. . ) o or ( . , .), and 1 1 . 1 1 will mean 1 1 . I (o - i.e., the t2 norm. These are similar to sequence spaces used in [17].

For points x E Z+ x Z, 1x1 denotes the t1 norm. If S : ZF", - RF", is a linear operator, we define a norm

Some further properties of these norms will be explained in Section 3.


The assumptions we have made on the nonlinear term g(x, u) in (2.1) imply conditions for the nonlinear operator W in (2.9), acting on the Hilbert spaces X u . In stating these estimates we fix an index o* < a and then make hypotheses that are required to hold for all o < (T* . For the nonlinear wave equation, the constant o is determined by the analyticity properties of g(x, u).

The basic property of the nonlinear operator W in the nonlinear lattice equation (2.9) is that it is an analytic mapping W : X u - Xu-yr for any o < o*. We use slightly stronger hypotheses, however, on the nonlinearity. Since we are interested in solutions of the nonlinear problem which are small perturbations of the normal modes of the linear problem, these properties are stated in terms of perturbations from a given linear mode. As usual, we concentrate on perturbations of the linear mode

(2.18)

-

1 cp(p)(x) = $PI + ip2)by(x) + (PI - iP2)by*(d) 7

for p = ( p l , p 2 ) E C2. We denote by Q the orthogonal projection in e2 onto the subspace spanned by cp(p). Under the symmetries of Section 2.3, the sequence cp(p) has the simple property that Tcq(p) = cp(Tcp), where Tcp = (cos([)pl - sin([)p2, sin([)pl ~ +cos([)p2). Additionally, if we denote the involution - (PI, -p2) = p* , then cp(p)(x) = cp(p*)(x) = cp(p)(x*). If p is real, then cp(p)(x) = cp(p)(x*), corresponding to a sequence which satisfies the reality condition. This is the lattice representation of the periodic solution (2.6) of the linear wave equation (2.3).

To state our hypotheses more precisely, define gU = {(x, [) E C2 I IImI < o,lIrn[I < o}. Given C > 0 fixed, let { $ n ( x ) } ~ o be any sequence of functions with the property

(2.19) sup I+Cln(x)I 5 Ce'lnl , n = 1,2,3,. . . . x:lImrl <u

For u E X u , set fib, [) = x j , k u( j , k)$,(x)eikc and define I I (uI I I c , ~ = sup{+n} S U ~ ( ~ , ~ ) ~ ~ ~ I i i (x, <) I, where the sup{,,,} runs over all families of functions satisfying (2.19).

Throughout the following statement of Hl-H3, we assume that u E X u and that rnax(llu)l,, I I IuI I I c , ~ ) 5 ro < 1 for some fixed C. Assume further that Qu = 0.

H1 For any 0 < y 5 o, W(q(p) + u) E Xu-y , and

(2.20)

H2 For any 0 < y 5 (T, D,W(cp(p) + u ) is a bounded operator on whose norm is bounded by

(2.21)


H3 For any 0 < y 5 0, the second derivative DtW(cp(p) + u ) satisfies the estimates

cw (2.22) IIDZW(dP) + u)[v>wlIl.-y 5 3 IlvllollWlln-y . Y

Furthermore this bilinear operator can be written in the form DtW = C + A. where A satisfies

(2.23)

The operator C = D:W(O) satisfies the property

for any z , ;’ with ( z - z’) = (0,O) or (+ 1,O). The operator C in (2.24) depends only upon the coefficient g2(x). In fact if

DtW(0) = 0, that is, if the nonlinear term is of cubic or higher order at zero, then C = 0. The role of hypotheses (2.23) and (2.24) is to avoid overly strong nonlinear coupling between resonances associated with lattice sites within the same singular region. They are used in Section 4 in the analysis of the linearized operators D,W and their dependence on parameters. In the above estimates, the constant CW is of course independent of y .

We emphasize that hypotheses HI-H3 always hold for the nonlinear term in (2.9) if the lattice problem comes from the nonlinear wave equations that we are considering, a fact we verify in Section 7.

We next discuss a second requirement placed on W - the twist condition, or condition of genuine nonlinearity. Let Bo = {x E Z’ X Z I ljl + (k l 5 &} be a bounded subdomain of the lattice Z+ x Z and define no to be orthogonal projection onto (’(Bo). This projection commutes with T , and, furthermore, no commutes with the lattice involution (2.13) and thus preserves the reality condition. Consider the approximate problem to (2.9) for a sequence UO E e2(Bo),

If the sequence {wj>,”,, is (do,&) nonresonant then lloV(R) restricted to e2(Bo) has a zero eigenvalue when R = W I , with eigenvectors supported on the lattice sites N = ((1, +l)}. There are no other singular sites in Bo. Parametrize a neighborhood of zero in e 2 ( N ) , the null space of n o V ( q ) by

Then Q is the orthogonal projection onto P2(N). Define its complement P =

(1 - Q).


Equation (2.25) possesses a branch of nontrivial solutions (Uo(p), Ro(p)) bifurcating from (p, R) = (0, w,). Since Tpp(p) = cp(Tcp), the above branch of solutions is T , invariant. The condition of genuine nonlinearity is that for this branch of approximate solutions the frequency parameter Ro is nondegenerate in p.

DEFINITION 2.6. The problem (2.9) is said to satisfy a twist condition if the bifurcation surface of the approximate problem %'o = (p, Ro(p)) satisfies

(2.26)

Since the branch of solutions is invariant under T I , then a,Ro(O) = 0 automatically.

In contrast to hypotheses Hl-H3, the twist condition is not always satisfied. Furthermore, in order to estimate the set of frequencies that remain when we complete our construction, it is necessary to have some lower bound on the deter- minant in (2.26). The next proposition, which we prove in Section 7, guarantees that such estimates do hold for most nonlinear wave equations. We fix < 77 < 1, a n d O < u < 1-77.

PROPOSITION 2.7. Choose L* > 0. There exists an open and dense set of nonlinearities such that for some h > L * , the nonlinear term W in (2.9) derived from a nonlinear wave equation whose nonlinearity is in this set satisfies the twist condition with

I det(d&(O))l 2 h' > 0 . We can now state the main existence theorem for periodic solutions for non-

linear lattice problems (2.9).

THEOREM 2.8. Consider equations (2.9) which satisfy the reality and translation invariance conditions (2.14), (2.15), (2.16). There exists a constant L , =

L,(Cw, v , r , wl), such that if the nonlinear term W in (2.9) satisfies hypotheses Hl-H3 with constant CW, and ifthere exists 2 L , such that:

(i) {w,>j"=l is (dO,b)-nonresonant with w1 for some do 2 Loq, and (ii) the twist condition holds,

I det(d&,(O))l 2 LO'' > 0 ,

then there exist uncountably many solutions U(x) E Xpc~/2 of (2.9). More precisely, there exists r* > 0, a Cantor set %' C [0, r * ) , and smooth functions (U(x; r ) E 2YT/2, R(r)) defined for r E [0, r*) , such that (U(x; r) , R(r ) ) is a solution of (2.9) for r E %'. Furthermore,

(2.27)


These sequences remain solutions when acted upon by the translation T, ; they are embedded circles in the space X Z , ~ .

Both Theorem 2.1 and Theorem 2.2 follow from Proposition 2.5, Proposition 2.7, and this theorem. Indeed, consider solutions to the nonlinear problem (2.1) described in terms of the eigenfunctions of the linearized equations. One expands a function u(x, t ) which is 27r/R periodic in time, satisfying the correct spatial boundary conditions (Dirichlet on [O, K ] , or periodic), in terms of the eigenfunction expansion. Let < = Rr, then

Square integrable time periodic solutions u to the wave equation (2.1) correspond to sequences U ( j , k ) E e2(Z+ X Z) which solve the lattice equation (2.9). The nonlinear term in the wave equation is g(x, u) - gl (x)u. This corresponds to the nonlinearity for the lattice system

We call equation (2.9) the mode interaction equation for the nonlinear wave equation (2.1). If U ( j , k ) E Z, then the solution u(x, <) given by (2.28) is analytic. If U ( j , - k ) = U ( ( j , k ) * ) = u(x), then u(x,<) is real, and vice versa. Similarly, one readily verifies that (2.15), and (2.16) hold for lattice problems coming from nonlinear wave equations. Thus it remains only to check conditions Hl-H3 on W, and the nonresonance and twist conditions. Proposition 2.5 and Proposition 2.7, however, guarantee that for open dense sets of nonlinearities g(x, u ) in (2.1) these hypotheses are satisfied. Thus, Theorem 2.1 and Theorem 2.2 follow.

Starting from a solution U ( j , k ) E Z, of the lattice equation (2.91, consider the function u(x, () = C(,,k)EL+ xz U ( j , k)+,(x)eik( and its translates by TO. Since U E X , they form an analytic family, in fact an embedded circle. Setting = Rt, a real analytic solution of the nonlinear wave equation (2.1) is obtained.

Our method of proof of Theorem 2.8 follows the Lyapunov-Schmidt procedure of bifurcation theory using the projection operators Q and P to break the lattice problem (2.9) into a finite dimensional piece and an infinite dimensional piece which are studied independently. Equation (2.9) is equivalent to the following pair of equations:

(2.30) Q(W(U) + V(R)U) = 0 P(W(U) + V(R)U) = 0 .

Accordingly, we decompose U = q ( p ) + u, where q ( p ) is in the range of Q and u is in the range of P. The following theorem contains somewhat more detailed information than Theorem 2.8 and, in addition, emphasizes the different


hypotheses which are necessary to solve the first and second of the equations in (2.30).

THEOREM 2.9. Consider equations (2.9) which satisfy the reality and translation invariance conditions (2.14), (2.15), (2.16). Let L, be as in Theorem 2.8 and assume that for some LO > L , the sequence { w j } p l is ( d o , h ) nonresonant with w1 with do 2 Go. (i) I f the nonlinear term in (2.9) satisfies hypotheses Hl-H3, then there exists

r* > 0 such that on the neighborhood JV* = {(p, 0); 1 1 pII < r * , 1 0- w1 I < r* } in the parameter space there is afunction u(x; p , 0) E X Z / ~ , with Qu = 0, which is C" in the parameters ( p , 0) E X* , and a Cantor set X E X, in parameter space, invariant under T,, such that for (p, 0) E N, u(x; p , 0) is a solution of the first bifurcation equation

2

(2.31) P(W(Cp(p) + u) + V ( 0 ) u ) = 0 ,

(ii) I f in addition to the above hypotheses the nonlinearity satisfies a twist condition

then there exists a C" surface 9' = ( p , R(p)) invariant under Tc satisfying the second bifurcation equation

where the intersection { p ; Y ( p ) f l X # 0 ) has positive measure.

The intersection of Y with X is of course the solution set for equation (2.9), consisting typically of a Cantor set foliated by invariant circles. The measure of the set { p ; Y ( p ) n .N # 0) is relatively large, on the order of rr?.

Note that solving the P equation does not fix any relationship between the frequency of the solution and its amplitude. This occurs only when one solves the Q equation and is dependent on the twist condition. This is typical of the Lyapunov-Schmidt procedure, and replaces the frequency mapping that is part of the classical KAM techniques.

Even if the twist condition fails to hold, one may have periodic solutions. Such situations are explored in [6].

It is possible that the actual bifurcation point ( 0 , w l ) is not an accumulation point of X due to an exact or near resonance of w1 and wj , with j such that ( j , k ) 6 Bo. If none of these occur, however, then there is a result on the density of the periodic orbits within radii 0 < llpll < r,. Fix exponents 0 < 7 and 0 < CU such that E + 1 > 7.


DEFINITION 2.10. A (do, h)-nonresonant sequence {wl}j”= I is fully nonresonant with W I if there exist positive constants C I and c2 such that for all ( j . k ) E Z+ x Z, ( j , k ) f (0.01,

and if for ( j , k ) # (1, T l ) ,

THEOREM 2.1 1. If the frequency sequence {w,} isf i l ly nonresonant and in addition the hypotheses of Theorem 2.9 hold, then the Cantor set of Theorem 2.9 has p = 0 as an accumulation point. Furthermore, there is an estimate of the density of periodic orbits near p = 0. There are constants p > 0, C, such that for all 0 < rl < r * ,

meas{r E (O,rl) I IIpII = r, (p,R(p)) E A‘} Z- rl ( l - C,rY) .

Because it depends upon the details of the induction process, the proof of this density result is deferred to Section 7. In the proof, estimates of the size of the exponent p will be given.

2.5.

Principal examples of problems of the form (2.1) are the nonlinear Klein-

The Nonlinear Klein-Gordon and sine-Gordon Equations

Gordon equation

(2.34) # u = a.:u - m2u + (m2/3)u3

and the related sine-Gordon equation

(2.35) afu = d.:u - m2 sin(u) .

Equations (2.34) and (2.35) both have frequency sequences,

for the Dirichlet problem and


for the periodic problem. (The term [ ( j + 1)/2] inside the square root denotes the integer part of ( j + 1)/2.)

When posed with periodic boundary conditions on the interval [O, 2x1, (2.35) is a completely integrable Hamiltonian system. This is not the case for (2.34), or for (2.35) with Dirichlet conditions posed at x = 0 , ~ . Traveling wave solutions for (2.34) and (2.35) satisfying periodic boundary conditions on the interval [O,T] are easily described using phase plane analysis for functions u(x - ct). Furthermore, time periodic solutions of more general form which satisfy periodic boundary conditions do not satisfy our nonresonance conditions. Resonant solutions are addressed in another publication; see [7]. We shall concentrate on the case of Dirichlet boundary conditions. On a formal level this problem was discussed by J. B. Keller and L. Ting in [I 11, and the curvature of the solution surfaces a$(O) was derived. These solutions are related to a class called “breather solutions” in the literature, which are spatially localized, time periodic solutions to nonlinear wave equations posed on all of x E R. Theorem 2.2 implies the existence of small amplitude time periodic solutions of both (2.34) and (2.35) for a set of values of the parameter m2 of full measure. We have only to check that the hypotheses of Theorem 2.9 are satisfied for the associated lattice problem. Either of these equations could be perturbed by an additional nonlinear term, h(x, u), and, as long as the twist condition is satisfied, the results discussed below would still hold.

THEOREM 2.12. For any sequence of constants d t ’ , L t ) such that

lim dt’ log($’) = 0 , n-m

there is an open set A of f u l l Lebesgue measure such that if m2 E A, then the frequency sequences (wj>,”=, for the equations (2.34) and (2.35) are ( d t ’ , L t ) ) nonresonant with w1 for some n.

Proof Fix do, Lo and consider an arbitrary interval [a, b] of parameters m2. For the Dirichlet problem, the first (do, Lo)-nonresonance condition is violated for those m2 such that lk2(1 + m2) - ( j 2 + m2)I 5 do. That is

hence by excising a closed interval of length 2do/ ( k 2 - 11 about every point ( j 2 - k 2 ) / ( k 2 - I) , l j l + Ikl I Lo, ( j , k ) # (1, ?l) which falls within the intervaI [a, 61, the remaining values of m2 satisfy the first condition of Definition 2.4. Note that this imposes no condition for k = t l , and that m2 = 0 is excised. The diophantine condition of Definition 2.4 is violated for those values of m2 such that I k - j I 5 do/( I j I + I k I)‘. Excising a closed interval of length 2&/( 1 j 1 + Ik I )‘ about every point m2 = ( j / k ) 2 - 1 as well, the remaining parameters satisfy both conditions in Definition 2.4. Call this open set A(&, Lo). The only ( j , k ) that

PERIODIC SOLUTIONS OF NONLINEAR WAVE EQUATIONS I427

need to be considered are those for which (1 + a)k? - Co 5 j' 5 ( 1 + b)k' + Co. The total measure of the excised intervals is estimated by

as long as T 2 2. The set A n [a, b] = U , , A ( d t ' , $I), thus if d t ' log($') - 0, the set "fl is of full measure. The proof in the case of periodic boundary conditions is similar.

Computing the curvature of the approximate bifurcation branches for the equation (2.12) is a straightforward exercise, and it is non-zero and independent of the choice of approximate domain Bo as long as Lo 2 6.

PRoPosrTIoN 2.13. Consider the equations (2.34) and (2.35), sarishing Dirichlet boundary conditions on the interval [0,7r]. The curvature of the branch bifurcating from the frequency wl is

(2.36)

In particular; I det(d;R(O))I 2 Lo' i f& is large.

Hypotheses Hl-H3 hold in this situation, so we can apply Theorem 2.9 to conclude

THEOREM 2.14. Suppose that m2 E A gives rise to a (do,&) nonresonant frequency sequence with & 2 L, and do 2 bPy. Then there exist small periodic solutions of (2.34) and (2.35) with frequency near W I .

If the sign of the nonlinear term in the equation (2.34) is reversed, that is, if g(x, u) = m2(u + ( 1/3)u3) then the curvature (2.36) reverses sign, but still retains a twist and the existence theorem holds.

3. The Induction Argument

3.1. Basic Estimates

In this subsection we introduce notation relating to the function spaces in which we solve (2.9). Using the family of Hilbert spaces X r , we show that the solutions of these lattice problems decay exponentially.

The following proposition shows that the operator norms defined in Section 2.4 respect the exponential weights, and clarifies their relationship to the usual operator norm, which we denote llS\l?.


PROPOSITION 3.1. I f S and T are linear operators from Xu to Xu, then

(i.e., 1 1 . I J u is a Banach algebra norm). Furthermore, if IISllu S CS, then there is a sup-norm estimate of the matrix elements of S ;

Iffor some o, the matrix elements of a linear operator S are bounded by

then for all 0 S y < o II s Ilu-ys 4Cs/y 2

l141:p s IlSllU . Finally,

The proof of this proposition is straightforward. An additional well-known and useful property of these norms is embodied in the following lemma.

LEMMA 3.2. Let B, = { ( j , k ) E Z+ X Z; l j l + Ikl I Lm}, and let I l m be the orthogonal projection onto e2(B,), with 1 - ll, its orthogonal complement. I f 0 < y S CT then

I I ~ m f IIU 5 eyLm Ilf Ilu-r

II(1 - I~ , , , )~ I I~ -~ 5 e-YLm I l f 110 .

Proof From the definition of the norm,

We also use the analyticity of these solutions as functions of various parameters. For this purpose, if JV C R2 x R, we define complex domains D ( N ; p ) =

{ ( z , ~ ) E ~2 x ~ ; J l z - pi12 + 101 - ~ 1 1 2 < p for some (p, 01) E N}.


3.2.

The induction procedure of this paper is started by solving an approximate bifurcation problem (2.25), consisting of the full equation (2.9), restricted to the lattice subdomain Bo. Let W O = HOW, Vo = noV, where no is defined prior to (2.25). As we have indicated above, the lattice problem will be solved using a Lyapunov-Schmidt decomposition;

(3.1)

The Bifurcation Problem on Bo

P(Wo(p(p) + u) + V,(R)u) = 0 ,

where Q is the orthogonal projection onto e 2 ( N ) and P = (1 - Q). This pair of equations is equivalent to (2.25).

With these definitions in hand, we can construct the first approximation to the solution of (2.9). For any subset B C Z+ X Z, write = B\N, where N = ((1, l), (1, -1)) is the set of lattice points supporting the null space of VO(WI). Define N O = {(p,O) E Iw2 x Iw 1 I IpI 1 < ro, IR - w1 I < ri} , a neigborhood of the point (0, ~ 1 ) at which bifurcation branches are to be constructed.

LEMMA 3.3. Suppose that the sequence { w j } F l is ( d o , h ) nonresonant with W I . There exist constants C > 0 and c > 0 such that if 0 < ro < d o / ( c G ) and ro = PO, then there exists a solution UO(X; p, 0) E e2(Bo) of (3.1) which is analytic in D(N0, PO). Furthermore,

and uo satisfies the estimate

(3.3)

for any u 5 u* - (1/h). For (p, R) E No real, uo satisfies the reality condition. For (p, R) in the complex subdomain D(N0, p0/2) the Cauchy estimates imply that

(3.4)

and in general for 1c-i 2 2,


Proof Given any lattice region A, we define HA(R; U ) to be the linearization of the function F(U) = W ( U ) + V(R)U about U restricted to A, i.e.,

We may suppress various indices of HA if they are clear from the context. Thus if we linearize the function on the left-hand side of (3.1) about uo = 0 and set (p, R) = (0, w1) we find

H,,(Wl) = PVO(W1) . By hypothesis this is invertible, with inverse G ~ ( w 1 ) bounded by C/do. Equation (3.1) is finite dimensional, and hence by the implicit function theorem it has a solution which is analytic in a complex neighborhood of (p, R) = (0, w1). The symmetry of the equation with respect to T , and with respect to complex conju- gation assures the covariance properties of the solution. The point of this lemma is to estimate the size of this neighborhood. Solutions of (3.1) are fixed points of the mapping

Unique analytic solutions of (3.1) are assured for parameter values such that the mapping in (3.5) preserves a neighborhood of the origin on which it is a contraction. To find a neighborhood that is mapped to itself, we use hypothesis H1 to estimate

Using Lemma 3.2,

and by optimizing over y such that o + y 5 o* we find that y = 1/Lo and


We require ro = po < do/CL& and define C, = CLo/do. For (p, 0) E D(”0; PO), the complex neighborhood of No, if lluol10 5 C,IIpI12 then llul [ I n < C,ll~11~ also. Thus the fixed point of the map will lie in this neighborhood.

We next show that (3.5) admits a contraction estimate. Consider differences of the R.H.S. evaluated at two points U I and 112. The difference of the first terms is

using Proposition 3.1 and hypothesis H2. In order that this gives a contraction mappin when llpll < ro + P O and llul l l n , l l u ~ l l ~ < C ‘ u ~ ~ p ~ ~ 2 , we ask that po = ro < (do/CL,). B

The difference of the second terms is easier,

which gives a contracting estimate if (R - W I I < r i + PO, and C(r i + P O ) < d o / G ; again, this follows from the hypotheses of the lemma.

Estimate (3.6) leads to an a priori estimate on the fixed point,

This is the upper bound stated in (3.3).

Over the neighborhood D(N0; PO) the problem (2.25) is reduced to finding the zero set of the mapping of (3.2); (p, 0) - Gdp, a) = Q(Wo((p(p) + udp, R)) +Vo(fl)(p(p)) E e2(N). A trivial solution branch is {(p,R) I p = 0, - r i < ( L U ~ - fl) < r i } , and we seek an additional family of solutions parametrized by {p 1 I(p(1 < ro}. As the equations (3.1) and (3.2) and the solution uo(x;p,R) respect the translation T,, the zero set of (3.2) is invariant under T,. Properties of the mapping are given in the next result.

LEMMA 3.4. The mapping Go(p,fl) is analytic on D(No,po), zero for {(p. 0) I p = 0, -ro < (wi - R) < r i } , and has a Taylor expansion at p = 0, 0 = w1 with the following Taylor coeficients:

2

(3.7)

1432

and

W. CRAIG AND C. E. WAYNE

If w1 f 0, the last term is nonvanishing.

Proof The analyticity follows immediately since the implicit function theorem guarantees that uo(p, 0) is analytic and p(p) is analytic by construction. We next compute the Taylor expansion of the function Go at (p, 0) = (0, WI). Clearly Q (Wo(cp(p) + uo(p, 0)) + Vo(R)cp(p)) I(p,n)=(o,w,) = 0, and the first derivatives are

where we use that cp(0) = 0 and d,,cp(O) is in the null space of Vo(w1). To compute the second derivatives we use that D,Wo(O) = 0, lluoll0 5 C,ll~11~, and the fact that (3.2) is covariant with respect to T,. Indeed,

Thus, Go is odd in p, and both d;Go(O,wl) = 0 and diGo(0,wl) = 0. The mixed partial is nonzero, however, as long as w1 # 0.

(3.10)

The zero set of Go(p, R) is described by simple bifurcation theory in the presence of the Tc symmetry. By the properties in Section 2.3, Go(p,n)(l, -1) = Go(p, R)(1, l), so it suffices to study the zero set of Go(p, O)(l, 1). Applying the translational symmetry, we find that T,Go(p, R) = Go(T,p, O), so that the zero set is Tc invariant. Using (3.7) and (3.8) of Lemma 3.4, an application of the Morse lemma implies that the zero set of Go(p,R) in a complex neighborhood of (p,R) = ( 0 , ~ ) consists of the R axis, union a surface (p,Oo(p)) given as


a graph over a neighborhood of p = 0 in @’. If we now use the facts that Go(p.R) = G o ( F , - a) and Ro(T,p) = flo(p) we see that whenever p is real, then so is Ro(p).

The T , invariance of the zero set of Go(p,R) also implies that the surface (p, Ro(p)) obeys dp,QdO) = 0, that is, it is tangent to the plane {(p, R);Q = wl}. By a further Taylor expansion we compute a;,,,Ro(O), which pertains to the twist condition of the nonlinear problem.

LEMMA 3.5. Under the hypotheses of Lemma 3.3 there exists a small positive constant c such that if L Q J ~ ~ < cdo, and G ( r i + p i ) << c(dopo), the solution surface (p, flo(p)) is dejned over the f i l l neighborhood {p E C’ I IIp - P I II < p o for some PI E R’ with JIpl 1 1 < ro}. The suface satisfies

(3.1 1)

where

If 730 > k”, then on the set {IIpII < rl = rAtE}, for E = (1 + 77 + v)/(4 + 277), the surface obeys the estimate:

flo(p) = W l + ~olIp1l2(1 + Y(p))

where IY(p)Ip < 1/2. This follows from (3.7), (3.8), and (3.1 l) , and the estimates (3.4) in the Taylor remainder.

Proof: The expression for the curvature is derived from the Taylor expansion of the mapping Go(p, a) at (0, W I ) . Differentiating (3.1) with respect to p, we easily find that

(3.13)

and di,,uo(O, W I ) satisfies


Since PVO(WI) has inverse GK,

d ; , p , ~ o ( O , ~ 1 ) = -GB~(w~)(D~WO(O)[~~, 'P, dpkcpI) .

In order to compute the curvature of the surface, the relevant third-order term in the Taylor expansion of the mapping Gdp, 0) is diGo(0, W I ) .

(3.14) d:Go(O, W I ) = Q((D3Wo(0)(dpcp)3) + 3(D2Wo(0)[dpcp, d&~])) .

Using the expression for dguo and (3.13) this gives (3.12) for the curvature of the nontrivial zero set (p,Ro(p)) of the mapping Go at p = 0. Incidentally, we may easily deduce that dndgGo(0, W I ) = 0, dAGo(0, W I ) = 0 since the mapping is odd.

We also establish that the surface (p, Ro(p)) is defined and analytic throughout the complex neighborhood {p I JIp-pl 1 ) < po for some p1 E R2 with J(p1II < ro}. Since the branch is simple it suffices to show that d,Ro(p) is bounded. Using that Go(p, Ro(p)) = 0 and differentiating with respect to p, we find

Q (Vo(R)dp, cp + dnVo(R)cpdP,

Thus

+ DWo(cp(p) + uo)(dP, cp + dp, uo + dnuod, Q)) = 0 .

Q(anVo(O)cp + (DWo)dnuo))dp,R = -Q(Vo(R)d,,cp + owodp,(cp + uo)) . Clearly IlQ(Vo(R)d,,cp + DWod,,(cp + u0))llo < C, so that the only possible singu- larities occur when Q(dnVo(R)cp + DWodnuo) vanishes. The first term is explicit - QdnVocp(p) = 2Rcp(p), which is bounded below by Cllpll for R bounded away from zero. Using Lemma 3.3, and a Cauchy estimate for the second term,

I1 QDWodnuo II o 5 Cw ( I1 P II + I I uo II n ) II dnuo I I

as long as constants are chosen as in the hypotheses of the lemma.

3.3. The Induction Hypotheses

Starting from the approximate solution of the previous subsection we will inductively construct better approximate solutions in ever larger regions of the lattice, and prove that in the limit they converge to a solution of the original problem (2.9). In this subsection, we state the inductive hypotheses, and in the next, we verify that they suffice to prove Theorem 2.9.

We begin the statement of the induction scheme with the definition of the constants which appear in the induction and a description of their role which we hope will help the readers orient themselves.


Inductive Constants

(1) At the n-th step in the iteration we solve a finite dimensional problem in a lattice box B,, = { ( j , k ) I l j l + IkJ S Ln}, whose size is given by

L n = 2 " h , n 2 - 1 .

(2) The amount by which our n-th approximate solution fails to be an exact solution is measured by

The small constant EO and the exponent K E (1,2) will be fixed when we discuss the convergence of the induction.

(3) As the iteration proceeds we encounter worse small denominators whose effects are estimated by

1

K" E , = E o , n E l .

6 " = E ' where a is a positive constant.

effects of a resonance 5,, can be felt. Set (4) There is a length scale Z, which estimates the distance over which the

B e,, = L,, ,

with p a small constant. In practice we shall be concerned with neighborhoods of singular regions Cun(S) = {x E Z' X Z : dist(x,S) < en}. We shall need to estimate the spectra of the local Hamiltonian operators Hs and HC,JS) = W(R) + D u W ) ( ~ f , ( s ) , which are defined on S and disks Ctn(S) respectively. Since L,, will be large, and p < 1 we shall have e,, << L,?.

( 5 ) The approximate solutions will decay exponentially in the size of the indices l ( j , k ) l at an exponential decay rate which will change during the iteration. This rate is determined by

Here, 00 < o* < where a is the width of the strip in which the nonlinear term in (2.1) is analytic. Furthermore, we choose 60 and o* so that for all n 2 0,

(6) The solutions will depend smoothly on parameters (such as the frequency fl and p, the amplitude and phase of the periodic solution). We require them to be analytic in a complex neighborhood, whose size is governed by

on > a/2.

&I

where po is the size of our original domain of analyticity (see Lemma 3.3).


(7) The sets N,, E N,-l . . . E NO are closed sets in parameter space over which we shall construct the n-th approximate solution.

Induction Hypotheses

We are now in a position to state the induction hypotheses. Equation (2.9) is equivalent to two equations obtained by the Lyapunov-Schmidt decomposition of the full lattice problem

(3.15) F(p, 0, u ) = P(W(Cp(p) + u ) + V(R)u) = 0

If the hypotheses of Theorem 2.9 hold, the constants a, p, T, and K may be chosen so that if &I is sufficiently large, there exist small constants EO and do = Lo7), positive constants CG, C, and a positive exponent y such that the following induction statements are true:

(n.1) There is a smooth function un(p, R;x) = u,-~(p, Qx) +vn-l(p, R;x) = u ~ ( p , R;x) + 1;:; v,(p, R;x) on NO x B,, analytic on O(N,, p,/2), which satisfies: (i) For any (P, E D(Jlrj+l, pj+1/2),

and on NO one has

(3.17)

For higher p derivatives there is a general estimate

(3.19)

This estimate quantifies our control of the Ck norms of v, with respect to parameters. Furthermore, v, is covariant with respect to the symmetry (2.15) and it obeys the reality condition (2.14).


(ii) For (p,R) E D(hf,,p,/2), the function u,,(x,p,Q) is an approximate solution of the first bifurcation equation (3.15). Specifically,

llF(ufl)llo" 5 llP1I2% '

(11.21 There exists a nested sequence of closed sets .C^O 3 .4'1 3 . . . A'",,+I, with the following properties: (i) If (p, a) E '4;+1, and if x, and xl are any two singular sites in B,+l\B,,

which are not in the same singular region, then the distance between x , and xJ is greater than 2t,+1.

(ii) If S is a singular region in B,+l\B,, the local Hamiltonian operators are not too singular. Specifically, for any (p,R) E D(N,T+~,p,+~),

dist(spec(Hdp, 0; u,)), 0) > & + I

dist(spec(Hc(,,-,(s)(p, a; u , A 0) > & + I

(iii) For any C" surface, n(p) = XJlpl12(l +Y"(IpII))+wl. with I X I Li' and IY(.)lp < 1/2 over {IIpII < r ~ } , then

1 +v meas { llpll E ( O , r l ) I (p ,Np)) E -f,,,+I> 2 rl - Cro .

(Recall that the exponent v was defined before Theorem 2.8.)

The convergence of an induction based on (n.1) and (n.2), which we verify in the next subsection, suffices to prove Theorem 2.9. The inductive estimates (3.17H3.19) imply in particular that throughout the parameter domain NO there is a bound on the derivatives of u, with respect to parameters,

Using the definition of the inductive constants, we can conclude that u, is smooth in ~4'0.

PROPOSITION 3.6. Suppose that

Then for all (p, Q ) E hp0.


and in general

(3.20)

Furthermore, un is analytic, and satisJies (3.20) for (p, R) E ~ ( J V , , , p,/4).

We verify in Section 6 that the hypothesis of this proposition is satisfied if we choose the inductive constants appropriately.

Remark. There is a related fact which we shall need in Section 7. Recall the 1 1 I . I I IC-,~ norm we defined in Section 2. Applying the estimates of induction hypothesis (n.l), we also find immediately that

PROPOSITION 3.7. I f

The induction hypotheses also allow us to prove the following important result about the inverse of the linearized operator which appears in Newton's method. We defined singular lattice sites in Section 2. A nonsingular region of the lattice is a connected set of lattice sites containing no singular sites. The importance of such regions is that the linear operator which occurs in Newton's method can be inverted by a simple Neumann series in such a region.

Given a lattice region E , we define the local Green's function

THEOREM 3.8. Suppose that the induction hypotheses (j.1) and (j.2) hold for j = 0, 1, . . . , n. There exists a choice of the inductive constants a, K, and 6, such that if LQ is suficiently - large, and A is a nonsingular lattice region, then for any domain E,+I C B,+1 U A, the Green's function on En+l is analytic on D(Nn+ 1, pn+ 1 ), and SatisJies

(3.21)

Under perturbations of u, a similar estimate holds. If


the following estimate holds,

(3.22)

We proceed now as follows. In the next subsection we demonstrate that Theo- rem 2.9 follows from the induction hypotheses. In Sections 4 and 5 we verify the induction hypotheses and prove Theorem 3.8. In these sections we demand that various relations hold between the inductive parameters. In Section 6 we prove that by an appropriate choice of the constants a , @, T , and K all these relations can be simultaneously satisfied.

3.1. Proof of Theorem 2.9

In this section we demonstrate that the hypotheses (n.1) and (n.2), n = 1,2,. . . , together imply Theorem 2.9. Set JV = nf12df l , the parameter domain for which we obtain a solution of the first bifurcation equation (3.15). The parameters -yfl governing loss of exponential decay in the induction process are chosen so that the decay rate at step n always satisfies on > a/2, for all n. Thus we have

PROPOSITION 3.9. Suppose

then u,(p, Q x ) converges to a sequence u(p, R;x) E X Z / ~ ,

which is C" as a function of (p, R) on the parameter domain N o . Furthermore, I l ~ l l z / ? 5 2CLo/do.

Proof This is immediate from the estimates on v, in induction hypothesis (n.1).

Induction hypothesis (n.l)(ii) implies that for (p, 0) E A" C NO the sequence ii satisfies the first bifurcation equation (3.15). In Section 6 we prove that the hypothesis of the lemma is satisfied if

To obtain a solution to the full nonlinear lattice problem, it remains to solve the second bifurcation equation (2.33). This is a finite dimensional problem, re- covering the zero set of the mapping G. Not surprisingly, we shall show that the solution is close to the approximate solution that was discussed in Section 3.2.

is sufficiently large.


LEMMA 3.10. Define the mapping

(P, 0) - G(p, 0) = Q(W(q(p) + u(p, R)) + V(R)q(p)) . The mapping G is C" on NO, zero for {(p,R);p = 0, -ri < (w1 - a) < r ; } and has a Taylor expansion at p = 0, R = w1 with the following Taylor coeflcients:

dpG(O, W I ) = 0 , dnG(0, W I ) = 0 ,

and

Proof The fact that G is C" follows from the fact that u is C" on No, which follows from the uniform bounds in (3.4) and (3.20). The Taylor coefficients are computed as in the proof of Lemma 3.4.

The zero set of the mapping G(p,R) is a union of the R-axis and a surface (p, R(p)) which is invariant under the translations T,, and is given as a graph over the neighborhood {p E R2 I llpll < ro}. This is the surface of solutions of the second bifurcation equation (2.33). It is close to the approximate solution surface (p,Ro(p)). Intersections of (p,R(p)) with N correspond to solutions of the first bifurcation equation as well, and thus are solutions of the full problem (2.9). Such intersections are guaranteed if the surface (p, R(p)) has nonzero curvature at the point (p, R) = (0, W I ) .

The surface obeys the estimate

R(p) = w1 + XllP1l2(1 + Y(PP)) ,

where on the set {pIllpll < r1 = rA+E},E = (1 + 7 + v)/(4 + 2771, we have IY(p)Ic2 < 1/2.


Proof The proof is essentially the same as that for Lemma 3.5, using the solution u in place of the approximate solution UO. The estimates (3.4) on uo from the bifurcation theory on Bo are augmented by (3.20) from the induction argument. The expression for the curvature can be rewritten as:

1 J f blk = yf 061k + -(a,, (p, (1 - no)D:w(o)[(a,, cp), (apt p), (apk cp)])

e 601

Since cp(p) E e ’ ( N ) , the two middle terms vanish. The difference is estimated using (3.20),

IJC - Xol 5 CIIDZWo(O)[a,cp,(d;u - a;uo)lIlo

(3.23)

Combining this estimate with the hypothesis of the lemma, we see that the curvature of the surface (p,fl(p)) satisfies 17fl > &”, and thus by the induction hypothesis (n.2)(iii) the set (p, fl(p)) n X is nonempty. In Section 6 we show that there is a choice of constants such that (3.23) holds.

4. Verification of the Induction Hypotheses

In this section we begin the verification that (n.1) and (n.2) lead to a valid induction. There are two main subsections; in the first, we show that if (j.1) and (j.2) hold for j = 0, 1, . . . , n - 1, then we can prove (n.l), and in the second we construct sets N,,+I which satisfy (n.2). Note that the function uo and domain No suffice to start the induction.

4.1. Verification of (n.1)

Assume that (j.1) and (j.2) hold for j = 0, 1,. . . , n - 1. Then (n.1) is a consequence of the following estimate on the Newton iteration.

PROPOSITION 4.1. There exists CG > 0 (independent of n) such that if the inductive parameters satisfy

(4.1)


and

(4.3) 4CLo(r0 + PO) 5 do ,

then there is a smooth function u,(p, R; z ) = uo(p, R; z ) + ~~i~ vj(p, R; z ) , defined on No X B,, which obeys the following estimates:

(1) For any j = 0, . . . , n - 1, v, obeys the reality condition (2.14) and the symmetry condition (2.15) is analytic on D(Nj+l; p,+1/2) and for any ( p , 0) in this set it satisfies

Furthermore, there exists a positive constant C such that vj is a C" jhc t ion of ( p , 0) on all of No, satisfying the following estimates.

and for la[ 2 2,

The usual multi-index notation for derivatives is used in this estimate. (2) For every ( p , R) E D(N,; p,/2), the function u, satisfies

(4.4) IIF(un(p,O))IIcn 5 IIpI12&n .

Proof The idea of this proposition is to construct vn-l by a variant of New- ton's method due to Nash and Moser. At the n-th iteration step we correct the expression ll,F(u,-~ +v,-l) as an approximation to zero, where ll, is the orthogonal projection onto e2(B,). Since we approximate

nnF(Un-1 + vn-1) a nn (F(un-1) + DF(un-l)vn-l) 9

a better approximation of a solution to the problem is given by performing an iteration step which entails inverting the linearized operator IIn(DF(un-1 ))-'lI,


= Ge;;(ur1-1). This Green's function is not defined on all of N o , but on the set D(J',. p f l ) it is boundedly invertible. On this set we define the modified Newton correction to url- I ,

From the estimates of Theorem 3.8, Ge;; is analytic on D ( N r l ; p n ) , and satisfies (3.22). Furthermore, by ( n - l.l)(ii), we know that I ~ ~ F ( U , - I ( ~ , ~ ) ) ~ ~ ~ ~ ~ ~ S

I I P I I ' E ~ - I , on D(Jrfl-~,pr,-1/2). Thus, on D ( N , , ; p , ) , GB;; and l-I f lF(uf,-~) are analytic, and

Furthermore, cf1-1 obeys the symmetry and reality conditions (2.14) and (2.15). Using (4.4) and (3.22) to estimate lIGe;;IIun~l-,,~l we have proved

LEMMA 4.2. domain it satisfies

ijn-l(p, 0;x) is analytic on D ( N n : p n ) , andfor any (p, fl) in this

(4.5)

We now construct vn-l by smoothly extending G,-l(p, 0;x) to all of "40. This is done essentially by setting ijrI- l = 0 on No\Nn, however we use a smooth cutoff function to obtain a C" function. The following result is from the appendix of [51.

LEMMA 4.3. For every R > 0 and for every compact set A C Cd, there exists a x E CO(Cd) n Cx(Rd) : Cd - [O, I] , such that x has support in YR(A) = U,,EA{T E Cd I 117 - voll 5 R ) and ~ ( 7 7 ) = 1 for every 7 E Y R / ~ ( A ) . Finally, for every positive integer k ,

Define xn-l to be a function satisfying the hypotheses of the lemma with d = 3, A = and R = p,/2, and set v,-1 = xn-13,-1. We can construct x to be rotationally symmetric with respect to the p variables. With this construction, vn-l obeys the reality and symmetry conditions (2.14) and (2.15).


LEMMA 4.4. Within the domain of analyticity D(Nfl;p,,/2) the Cauchy estimate applied to (4.5) controls all derivatives dnd:v,-l(p, 0; .). Using Lemma 4.3, the estimate on all of No is,

P

and for Ja yI 2 2,

Lemma 4.2 and Lemma 4.4, together with the remark that for dist((p, a), A',) 5 i p , , V,-I = ijn-lr imply the first assertion of Proposition 4.1.

We complete the proof of Proposition 4.1 by verifying that Un = un-l + vn-l

is an approximate solution of (3.15), the first bifurcation equation F(u) = 0. Hy- pothesis (4.2) guarantees that the norm satisfy IIu, l l u n - , - Y n - l 5 2CLollp1I2/do, and I I I lun-l-(3/2)yn-l 5 2CLollpIl2/do. Furthermore, since llpll 5 ro + P O , hypothesis (4.3) implies that m a x ~ l l u f l ~ ~ u n ~ l - y n ~ , , I I lunl I lun~l-(3/2)yn~l) 5 yo, so we can apply hypotheses Hl-H3 to W(cp(p) + u + n).

The nonlinear term can now be estimated.

F(u,) =F(un-l + vfl-1) = ((1 - QI)F(ufl) + nn(F(un-1 + vn-1) - (F(un-1) + DF(un-1bn-1)) 5

(4.6)

where we used the fact that v,-1 was constructed so that

(on OWfl; pfl /2) ) . We can bound the various terms in (4.6) by using the following observations:

(i) From hypotheses H2 and H3, there exists a positive constant, CW such that for any 0 < CT*, the nonlinear term satisfies IIDuW(u)IIu-y 5 Cw/y2 and IlD?iW(u)[v,wIll.-, 5 (Cw/y3) I I ~ l l ~ l l w I I ~ - ~ , for all u with IIuIIu 5 1.

(ii) Using the explicit form of V(0) , we see that IV(R)(j,k)( 5 ( 0 5 a x + C ) ( j 2 + k2) , thus

where a,,, = W I + dm, is the maximum value of 0.


The fundamental theorem of calculus implies that

completing the estimate of the truncation error. Using the Taylor remainder formula, we shall estimate [F(u,,-i + vn-l) -

F(u,-~)-DF(u,-~)v,-rl = so so D F(u,-l +svn-I)v,,-lv,-l dsdt. This is quadratic in ~ ~ - 1 . From the first of the observations above,

1 1 2

In deriving this estimate we used the fact that D$(u,-l) = D ~ W ( U , ~ - I ) . Combin- ing these three estimates, we obtain the following result.

LEMMA 4.5. Under the hypotheses of Proposition 4.1,

llF(4l)IlU" 5 IlPl12En 3

for ( p , Q) E D W n , pn/2 ) .

Proof The estimates above imply that the truncation error is

\ \ ( I - ~ l l ) ~ ( u r l ) ~ ~ u , t s e-Y,-ILn II F(un) II L7-l-27"- I


The contribution from the quadratic error is bounded by

The proposition follows if the sum of these two terms is less than E,. Com- bining these two estimates and using the fact that llp112 S 4r& this follows from the hypothesis of Proposition 4.1.

4.2. Verification of (n.2)

We continue now by showing how one constructs the set N,,+I, described in the induction hypothesis (n.2). We construct N,+I using the properties of u, given by (n.1). In Section 2.3, the initial neighborhood is defined to be NO = {(p,R) I IIRepll < ro, IIImplJ < ro, and IR - w1 I < r;}. The following proposition not only implies (n.2), but includes some other useful information as well.

PROPOSITION 4.6. There exist constants co,d, and C such that if No = [CO log(&)/p], and the following relationships hold between the inductive constants,

(4.7)

then there exists a closed set &",,+I G N, such that:

(a) If(p, 0) E N,+I, then any two singular sites in Bi, which are not in the same

(b) If S is a singular region in B,+ 1\B, and (p, 0) E D(N',+ I , pn+l) then singular region, are separated by a distance of at least 24,+1.

PERIODIC SOLUTIONS OF NONLINEAR WAVE EQU.4TlONS 1447

If the inductive constants satisfv

then .A ;,+I has positive measure. additionally,

(4.9)

and for some p > 0,

then the set A’,, 1 satisfies the following intersection condition: (c) Let R(p) = IJI + JGIJpJj’(1 + Y’(JlpII)) , define a surface with nun-degenerate

quadratic contact at p = 0. I f (XI > b-”, and on the set (0 5 llpll S r l } we have IY(.)lp 5 1/2, then the set (0 5 r < ro I r = IIpII, (p ,R(p ) ) E J l r r , + l }

has measure greater than rl - Cro 1 +P .

In fact the set Jlr,,+, is invariant under the rotations p - T,p. This invariance is a consequence of the invariance of the spectrum of the linearized operator f f ~ ~ ~ , (u). under the rotations u - T,u.

LEMMA 4.7. The spectrum of the operator HE(T,u) is independent of <.

Proof: The properties of the nonlinear function in (2.9) are that T<F(u) = F(T:u) , and furthermore T E commutes with orthogonal projection onto t 2 ( E ) for any E C Z+ X Z. Differentiating with respect to u we find T,DF(u)v = DF(T:u)T:v. thus DF(u) and D F ( T p ) are unitarily equivalent, and the result follows.

Part (a) of Proposition 4.6 can be proven without restricting p - we need only place conditions on R. From the diophantine condition on LJI there is an integer N o such that for the initial induction steps 0 5 n < No, the separation condition (a) is satisfied for all ( p , R ) E .NO. This is the result of the next lemma, whose


hypotheses form a subset of those of Proposition 4.6. (We may assume without loss of generality that po and 5 are less than or equal to one, and that CW is greater than one.)

LEMMA 4.8. There exist constants co and C such that ifb is suficiently large, if the following relationships hold,

co log(2) < 1 - P ,

and if No = [cg log(b)/p], then for all n < No and (p, a) E J,, if x = ( k , j ) , x’ = (k ’ , j ’ ) are a pair of singular sites in Bi which are not in the same singular region, they are well separated; i.e.,

dist(x,x’) 2 2e,+1 .

Proof Assume that Ix - x’I < 2en+1. We derive a contradiction. If x = ( j , k ) is a singular site then Ij2a2 - W: I < d,. Factoring the left-hand side of this inequality and assuming that j and k are non-negative, this implies IjR - W k I < d,/ljR + wkl 5 C ( f l m a x ) d , / ( ~ j ~ + lk l ) , for some constant C(Rm,). Similar estimates hold for Ij’R - Wk’ 1 , and for the cases when j and j ’ are negative. We now use the asymptotics of the Sturm-Liouville operator. For the case of Dirichlet boundary conditions there exists a constant C,, such that Iwk - kl < C , , / k , for all k 2 1. (For a review of the properties of these eigenvalues see, for example, [18].) For periodic boundary conditions, if k = 2m or k = 2m - 1, m E 1, then I w ~ - 2m I < C,,/m. Since x and x’ are in Bi, there exists a constant C(Rmax) such that min(lkl, Ik’ 1 ) 2 C(Rma)-’L,. In the case of Dirichlet boundary conditions we have

I ( j - j’)R - ( k - k’) I = l(j - j’)R - (Wk - Wk’) f (Wk - wk’) - ( k - k’)l

(4.11) 2C(Rmax)ds + 2C(flmax)Cg,, C(gl,n)

I - s - . - Ln Ln Ln

Since x E Bg we have l j l + IkJ 2- L,, with a similar estimate for x’ . We are assuming that dist(x,x’) = I j - j ’ I + Ik - k’ 1 5 2e,+1. If n 5 No, 24,+1 5

< b, if colog(2) < 1 - P and the LO is sufficiently large. The principal frequency w1 is (do,h)-nonresonant, thus it satisfies a finite diophantine condition over the lattice points within Bo. Hence as long as 2e,+1 < b and IR - W I I < r;, then ( x - x ’ ) E Bo and the components satisfy

2NoP+1& = 2 h co log(2)+4

l ( j - j’)R - ( k - k’)l 2 I(j - j f ) W 1 - ( k - k’)l - l ( j - j’)(R - W1)l

(4.12)


By the hypotheses of the lemma, ~ C , , + I r i < ;d0/(2e,+l)~, so we find I ( j - j ’ )n - ( k - k’)l 2 ;do/(2Yn+1)‘. Combining this with (4.1 1) gives

4 (1-74) or. using [,,+I = L,+I, L,?+, d 2“’+”C(g1,R,,,)/do. This contradicts the hypothesis of the lemma if we take C = C(gI,Omax), and this contradiction estab- lishes the result.

In the case of periodic boundary conditions, the argument needs to be modified only slightly to account for the different asymptotics of the eigenvalues. The estimate in (4.1 1) is replaced by

where [ . I denotes the integer part. Combining this estimate with the lower bound coming from the fact that w1 is (do,&)-nonresonant, we find that dist(x,x‘) cannot be less than 2e,,+1 except in one special case. This special case occurs when k = 2m and k’ = 2m - 1 or vice versa. Then (4.13) becomes I ( j - j’)fll 5 C(gI,flmax)/L,. But if, in addition, j = j ’ we cannot apply the diophantine estimate to obtain the contradiction. Thus, if ( k , j ) and (k’, j ’ ) are any pair of singular sites with either j # j’ or ( k , k ’ } # (2m - 1,2m}, then they are separated by a distance of at least 2C,+1.

Remark. As a corollary of the proof of this lemma, we see that in the case of Dirichlet boundary conditions, the singular regions consist of isolated singular sites, while in the case of periodic boundary conditions the singular regions consist either of isolated sites, or pairs of adjacent sites, with j = j ’ , and {k,k’} = (2m - 1,2m}.

Singular sites are not necessarily well separated for all (p, R) E NO, for induction steps n 2 N o . In order to insure that part (a) of Proposition 4.6 will hold when n 2 N o , we delete from (w1 - ri,wl + r i ) those values of R for which it would fail to be true. For this purpose we use the following elementary result in Diophantine approximation.

LEMMA 4.9. For any bounded interval of the real line, I , of length (11,

(4.14)


With these preliminaries we may start the inductive definition of N,+ I . For n 2 N O define

(1) This will delete finitely many open disks from A”!?, resulting in the set N,+, G

Nil) having at most (44n+1)2 many components. We assert that if d is chosen appropriately, and R E JV,+~, then any two singular sites x,x’ E Bi which are not in the same singular region satisfy dist(x,x’) 2 2e,+1. Indeed, the argument used in Lemma 4.8 can be modified to prove this degree of separation. The one change is that in the estimate leading to (4.12), if R E .N!i 1 , then Ijs2 - k I 2 d( I j I + I k I for all 0 < l j l + Ikl 5 4e,+1. Thus, (4.12) is replaced by

(1)

(4.15)

and we have obtained the following result.

LEMMA 4.10. Assume that the hypothesis of Lemma 4.8 holds. Then there exists C > 0 such that i f the constant d in (4.14) satisJies

then for (p, R) E JV!!~, any pair of singular sites in Bg, which are not in the same singular region satisfy

dist(x,x’) 2 2t,+l .

Since we define N,+l to be a subset of this lemma implies part (a) of Proposition 4.6.

Next consider part (b) of Proposition 4.6, where we must address the dependence of the local Hamiltonians and their spectra upon the parameters (p, 0). In order to belong to a singular region S , a lattice point x = ( j , k ) must satisfy (wl - ro)lkl - CO 5 l j l 5 (wl + ri)lkl + CO. Furthermore, in either of the above problems a singular region consists of either one site alone, or a pair of adjacent sites of the form XI = (2m - l , k ) , x2 = (2m,k). Suppose S E B,+I\B, is a set of this type so it has the possibility for being a singular region for some (p, R) E NO. The local Hamiltonian for the surrounding neighborhood Ce,,,, ( S ) is

2

HC&+,(S)(P, 0; u,) = (~uW(Cp(P) + u,) + V m ) ) ICC,+,(S) .

This matrix has at most 4ei+1 many eigenvalues, which we list in order; ei(p, R), for i = 1,2,. . . . In this subsection we study the behavior of the eigenvalues of Hs


and Hci, ,_,(s) with respect to parameters. The goal is to control the subsets of .NO on which either Hs or Hci, , ,(s) has a small eigenvalue, of size lei(p, R)l < 6,+1. (To save space, unless the meaning is unclear, we shall write Hc,"-,(s) as Hccs).)

LEMMA 4.1 1. Under the hypotheses of Proposition 4.6, if" S is fixed, each eigenvalue ej(p,R) of Hc(s) is continuous in ( p , R ) E NO, differentiable almost eveqwhere in D(.4:,: pn/2), and is strictly monotone decreasing in Re 0.

Proof: For S c B:. as long as (,+I < L,/(2wl) + CO, every lattice site .Y = ( j , k ) E C(S) satisfies k > L,,/2. Then the local Hamiltonian Hc(s) is monotone decreasing in Re 0. Indeed, let R be real, then

The second term is bounded by

For (Cw&ri/(a3dopo)) < 1 < Rmi,L,'/2, the quadratic form (4, Hc(sy,!~) is monotone decreasing in R. and thus for fixed p the eigenvalues ei(p, 0) are monotone decreasing in R.

Note that if there are no quadratic terms in the nonlinearity W , then

(4.17)

a better estimate than (4.16). The eigenvalues vary continuously with respect to ( p , 0) E NO, however they

are not necessarily smooth in NO. Examples of Rellich (see [21]), show that even a two by two matrix depending analytically on several parameters may not have differentiable eigenvalues. This lack of smoothness occurs at points of collision between eigenvalues. On sets, however, where either all eigenvalues are distinct, or the multiplicity of all eigenvalues does not change, it is possible to choose eigenvalues and eigenvectors to be smooth functions of ( p , 0); see [ 101. In D(Jc:,; p,/2) these sets on which the eigenvalues of Hc(s) have constancy multiplicity are semi- analytic sets. thus e,(p, 0) are differentiable almost everywhere on D(Nfl; p,/2), and the eigenvectors can be chosen differentiably almost everywhere as well.


Let us fix a set S which has the possibility of being singular for some (p, R) E NO, and assume that for x E C(S),k > 0. We are concerned with the location within NO of the zero sets Zi = {(p, R) E NO; ei(p, R) = 0). For p = 0 we know that ei(0,R) = -(R2m2 - w$) for some (4,m) E C(S) , so that it vanishes only for R = we/m. For each p E {IIpII < ro}, ei(p,R) is monotone decreasing in R, thus Zi n {p = const} consists of at most one point. By the continuity and strict monotonicity of ei(p,R), Zi is given by a surface (p,Ri(p)), where Ri(p) is continuous but not necessarily smooth, and Ri(0) = we/m for some (4, rn) E C(S). Using Lemma 4.7, the sets Zi are invariant under T,.

Small neighborhoods of these sets Zi contain all the parameter values in NO for which the local Hamiltonian Hc(s) has a small eigenvalue.

LEMMA 4.12. Let ei(p, R) be an eigenvalue of Hc(s) such that Jei(p, R)I < & + I . r f the hypotheses of Proposition 4.6 hold, then there exists C > 0 and 01 with IR - R I I < CS,,+1/L? such that ei(p,RI) = 0.

Thus by excising all (p, R) E NO with dist(R, 01) < CS,+I/L?, for any (p, 01) E Zi, i = 1,2,. . . , the remaining parameters avoid small eigenvalues for Hc(s). If this excision is performed for all regions S G B,,+l\B, which are potentially singular regions, the remaining set of parameters will satisfy (b) of Proposition 4.6. Our goal will be to estimate the measure, and the geometry of the remaining parameter region.

Proof For fixed p, ei(p, R) is a decreasing function of R, thus it is differentiable almost everywhere, and by Fatou’s lemma

for any R I 5 0 2 . Given ei(p, R) and associated eigenvector $i,

ei = ($i,Hc(s)$i) 9

thus where ei and $ i are differentiable,

The eigenvectors are normalized, so that

hence

(4.18)


This is known as the Feynman-Hellman formula. We can estimate the derivative,

- (4.19)

To obtain the subset N ~ ~ l G JV, C NO on which all local Hamiltonians avoid small eigenvalues, delete from JV, all points (p,R) such that there is some S C { ( j , k ) E B,+I\B,, I (w1 - ri)lkl - CO < j < (w1 + r i ) l k l + CO), and some (p, 01) E Z , a zero set for an eigenvalue of the operators H s or H C , ~ * , ( S ) such that 101 - 01 < CS,+I/L:. The remaining parameters in ( 2 ) will now satisfy

Iei(p,R)I > 2&+1

5 CL,?+,p,+, + cw - cT3 ( 1 + 2 - 2 + 22) ~ P n + l

as long as both

(4.20)

Note that the first of these follows if p o < Z4/2C, which we may assume to be true with loss of generality. The second follows from the hypotheses of Proposition 4.6.


The remaining parameters in D(Nfl+l, p,+l) will now satisfy (b) of Proposition 4.6. We can estimate the total measure of the region NO that is excised by this process. The number of possible singular regions S is bounded by CLfl(l +riLfl+l). For each region S and surrounding neighborhood Ce,+,(S) we excise C6,+1/L: a neighborhood of size of all zero sets Zi . There are one or two small eigenvalues for S and at most 44i+1 for Ce,+,(S). Each of these neighborhoods is invariant under T E , thus the total measure of NO that is excised is bounded by

( 1 ) (2) Define Nfl+ 1 = Nfl+ n Nn+ the good parameter region for the (n + 1 ),' induction step. Conditions (4.8) insure that the sum over n of the measure of these excisions will be small when compared to meas NO = 27~ri .

This is still not enough to prove statement (c) of Proposition 4.6, which is the guarantee that a smooth surface (p , R(p)) with nonzero curvature will intersect Nfl+l for a set of p of large measure. This will rely on more detailed knowledge of the geometry of the sets Z i .

LEMMA 4.13. There is a constant C > 0, such that ifCL&/Z3dopo < 1, then there exists a constant C1 such that for any zero set Z i , there exists (t, m ) E Ce,,, (S) such that Zi lies between the cones

Proof We have chosen ei(p, R) so that ei(0, we/m) = 0 for some x E (t, m ) E C(S). The operator Hc(s) is monotone decreasing along any line (p(s) ,R(s)) = (spo, we/m + sCl/L;) E No. Indeed, using the arguments of Lemma 4.12, we have

d c1

ds Lfl -Hc(s)(P(s), Ws); ufl) = + M C ( S ) + Po . dpHc(s) *

The p derivative is bounded by

Under the hypotheses of the lemma we can choose C1 such that the monotonicity of dnHc(s) dominates the variations given by p . dpHc(s). Thus the operator is monotone decreasing along (p(s), Ws)), and so are its eigenvalues.


The cone ( w t / m ) - (C,/L?)I/pIJ is shown to be a lower bound with a similar argument.

Remark. In the case where the nonlinear term W contains no quadratic terms, there is a better estimate of both &W from (4.17), and i3,W;

(4.22)

The sets Zi can then be shown to lie between the paraboloids

The more subtle control over the behavior of the sets Z , that we require is that within the region A ’ * ~ ~ l , there is a form of parabolic estimate even for the general nonlinearity.

LEMMA 4.14. Under the hyporheses of Proposition 4.6, we can choose C > 0 such rhar if CI = C(l/d, + 2(b/do)), and $(PI, Rl), (p2,%) E Zi also lie within the same component of N,,+l, then ( 1 )

(4.23) llP2Il2 - llPl 1 1 2 I

That is, the sets 2, are restricted by paraboloids rather than cones, whenever they intersect A”!: The proof of this lemma depends upon several facts. First, for (p, Q) E J ’ ” ~ ~ l there can be at most two eigenvalues of the local Hamiltonians Hs(p? 0; u,) and Hccs,(p, R; u,) which have smaller norm than 3dJ4. Certainly both domains S and C(S) have only one singular region, so they contain at most two singular sites at which IV(n)(x)l < d,. Then the operator norm of D,,W(cp(p) + u,) is bounded:


Thus if 16Cwro/a2 < ds, the operator norm of D,W is bounded by dJ4, and the eigenvalues of V(R) can be perturbed by at most this much by its inclusion.

Secondly, the eigenvectors associated with the small eigenvalues of Hc(s) have special structure. They are supported principally on the singular region S , with estimates of their size at other sites of C(S). Denote as usual Ps the orthogonal projection onto t 2 ( S ) . Let (i,hi, ei) be eigenvector-eigenvalue pairs for the eigenvalues such that I ei I < 3d,/4.

PROPOSITION 4.15. Under the hypotheses of Proposition 4.6, if (p, R) E

Proof This result is based upon Corollary 5.5 on the structure of the Green function for a region of the character of C(S) . There is a contour V surrounding all eigenvalues of Hc(s) with lei1 5 d,/2 such that dist(V,specHc(s)) > d,/16. For Cwl(p(l/F2 < Cd,, whenever 6 E V the hypotheses of Corollary 5.5 are satisfied, and we can represent Gc(s) = Go f~ Gs + R, where D = C(S)\S, Go is analytic over 151 < 3dJ4, and R(x,y) satisfies the estimate I(Rllo 5 Cllpll/d:.

Define the orthogonal projection PE related to the small eigenvalues

which has range in t 2 ( S ) . Since Go is analytic in a 3d,/4 neighborhood of the origin, -

P E = - f Go(<) fB Gs(<)dC 27ri v 1

When applied to an eigenvector i,hi of a small eigenvalue ei lying within the contour V,

(1 - PE)+i = +i - ~ 27ri 4 v Gc(s)+id() - ' 27ri f v R+id< (

1 - - --f R+id(. 27ri 4

Estimating the norm of the remainder,

where we have used that the contour has circumference proportional to d,.


For (p, R) E . N ~ ~ ~ l we are now prepared to derive better estimates on the behavior of the sets Z,. Keeping in mind the Feynman-Hellman formula (4.18) for perturbation of the eigenvalues ei(p, Q), we compute (+,, dPHc(s)(p, R; uf l )GI) and estimate its size.

PROPOSITION 4.16. Assume that the hypotheses of P roposition 4.6 hold. Then there exists a constant C > 0 such that ifC2 = C( l/d, + 2&/do)/Z2, and (p, R) E $4' there is an improved estimate

I($,, dpfk(s)(P,fl;ufl)+,)I = I(+I,D;w(p(p) + u,,)[dp((p + Ufl), $ I D

(4.24) 5 CZIIPII '

Proof The gradient with respect to p of Hc(s) is D;W(cp(p) + u,)(ap((p + un)). Using hypothesis H3 this is described (D:W((p + u, ) )~(s ) = (C + A),,,,. Using this along with the structure result for the eigenfunctions of Hc(s), ($i, D;W(cp(p) + u,)[dp((p + un),$i]) will be shown to be proportional to llpll.

Referring to H3, we find that (P&, C[dpcp, P&]) = 0, and we can estimate the remaining terms. The second term of (4.25) is

and the third term is even better behaved:

and finally, the most significant term is


where we have used the induction hypothesis (n.1) to control Ildp~,Ilon-y,. As- suming that llpll I rg < d, and collecting terms, the result follows.

This is enough information to prove the parabolic estimates on the sets Zi n Let (pl,Rl) E Zi f l N!il, and consider a path (p(s),fl-+(s)) = (PI +

s(pl/llpl II),Rl ? Cl(llp(s)l12 - ll~l1\~)/2L;). In the case of Dirichlet boundary conditions, if (p,R) E JV,+~ then the small eigenvalue el is isolated. Hence it is smooth with respect to parameters and

(1)

Choosing the plus sign to illustrate one case,

where (4.18), (4.19), and (4.24) are used to estimate the inner products. By choosing the constant C in Lemma 4.14 sufficiently large, we can insure that C152,in > C2 which implies that dei/ds S 0 along the curve (p(s), fl+(s)). Hence

and Zi must remain to the left of (p(s), fl+(s)) as long as this curve remains within A similar argument shows that Zi lies to the right of the curve (p(s), R-(s)).

With this knowledge we can prove Lemma 4.14. Let (pl,Rl) and ( ~ 2 , 0 2 1 E Zi, lying within the same component of Nn+lr with llp~II S IIp2II. Since Zi is invariant under T , we may also assume that p1 and p2 are parallel. Consider the curves (p(s), R,(s)), with p(t) = p2 for some t . If (p(t), R+(f)) E N:il then by our analysis

(1)

from which we conclude that

In particular, each set Zi can intersect a surface (p, R(p)) at most once in each component of Nn+l, the intersection being a circle in No. Recall that N!il consists of open disks that are constant in the p direction, thus if either of (p(t), fl,(t)) is not

( 1 )


within A”:;ll, then automatically (4.26) holds. This finishes the proof of Lemma 4.14 in the case of Dirichlet boundary conditions.

In the case of periodic boundary conditions, when(p, 0) E NriI.there may be two small eigenvalues, and the pairs (e,(p, a), cp,(p, R)) may not be smooth, due to problems with eigenvalue multiplicity. Nevertheless, the estimate (4.26) still holds. Indeed, if P d denotes the projection onto the space spanned by the eigenvectors Q1,$2, then the quantity (1/2)(el + e2) = (1/2)trPdH~(~#’,j is smooth, and on sets on which el = e2 its derivative replaces @/&)el in the proof of (4.26). This finishes the proof of Lemma 4.14.

We now are in position to prove part (c) of Proposition 4.6. Consider an arbitrary C” surface V = (p,O(p)) which is invariant under T,; R(p) = W I + Xllp112(1 + IIpllY(Ilpll)), where X # 0. We shall see that this surface intersects the good set of parameters N,,+, = A”:l, n NFl, in a large set. In practice these surfaces are solution sets of the second bifurcation equation (3.16), and intersections with nzoNfl correspond to solutions of the full equations (2.9).

The total measure of fl within {q - ri < fl < w1 + r i } which has been excised in order to construct X:ll from Xk’) is estimated in Lemma 4.9 by Cd/eA( 1 + r i tn ) . From the parabolic nature of V an estimate of the new excisions is given by,

meas(0 5 r < ro I (p, fl(p)) E NL’)\(N!!!l) with llpll = r ) (4.27)

The parabolic estimates in Lemma 4.14 are used to control the effect of the excisions in order to satisfy condition (b) on the surface +?. Suppose that (PI, Npi)) €

V n Zi 17 A‘;! 1 , and let (p, 0) E V be another point in N!lI, with I R - 01 1 < b,+~/Li. We have an estimate on the quantity lllpll2 - I(p11121. Without loss of generality consider p parallel to P I . On Z; fl N!?I, IQi(p) - fli(Pl)I 5 ~ ~ ~ / L ? ~ l l l p l l ~ - IIp111~1, while on g, IfHp) - W p d > IJC/2lI IIpII’ - llpi1121. Thus in forming NfJI by deletion of 6,+1/c neighborhoods of each Zi, any other points of NO that are excised satisfy

As long as C I / L ~ < IJC1/4, we have lllp11* - IIpll121 < (4/\XI)hfl+l/L;. Elim- inating such p for each possible singular site, each resulting eigenvalue set 2; and each possible intersection of Zi with the curve V, the total measure of the parameters p excluded in constructing N\:ll from A:, is controlled by


5. The Green’s Function

The principal goals of this section are to prove estimates on solutions to the linearized first bifurcation equation (2.3 l), where the linearization is about an approximate solution obtained in the induction. Actually, solutions are obtained for the linearized equations restricted to finite subdomains t 2 ( B , ) of t 2 ( Z + X Z); this restriction is both the analog of the smoothing procedure of the classical Nash- Moser method, and it allows us to consider only finitely many possibly singular sites at each induction step. In the induction the domains B, G B,+I on which we approximately solve (2.31) grow. In addition, the tolerance for small denominators, or small spectra, increases, quantified by the sequence 6,. The full nonlinear equations (2.9) also have small or zero eigenvalues, stemming from the null space of the operator HB,(O, w 1); the Lyapounov-Schmidt decomposition, however, allows us to work independently of these small eigenvalues. In the induction the linearized operators HE,, and the Green’s functions GB, depend upon the parameters in the problem, the frequency R, and the parametrization p of the null space of HB,(O, WI). The first step of the induction defines GB~(P, R) for (p, R) E NO a full neighborhood of the bifurcation point (0, wl). In subsequent induction steps we are only able to retain control of Gg,(p, R) in subsets N, G N,- I G . . . No, on which the tolerance of the n-th linearized operator HB,,(P, R, u,) for small spectra is satisfied.

5.1.

The first bounds on the Green’s function concern the nonsingular case. We denote N = {nj} the lattice sites which support the null vectors of HB~(O, W I ) and Bo = Bo - N. For (p,R) in a neighborhood of ( 0 , w l ) the diagonal elements IV(R)(nj,nj)l can be very small. Since we assumed that our initial frequencies were (do,&) nonrenonant, however, all of the sites of EO have IV(wl)(x,x)l > do. A lattice site x E Z+ x Z is called singular, if IV(R)(x,x)l < d,. When restricted to nonsingular sites, there is a go such that the operator H(p,R) is invertible on x, for all 0 5 o < 00. This is the conclusion of the following.

The Green’s Function on Nonsingular Domains

-

THEOREM 5.1. Let HA = V ( 0 ) + DW be a linear operator defined on t 2 (A) , for any domain A c Z+ x Z consisting entirely of sites where IV(x) I > 6. Suppose that DW(x, y ) satisfies an estimate of the form

(5.1)

Then for

and for all IzI 5 i 6 the Green’sfunction GA(z) = (HA - ~ 3 - l exists and decays exponentially ofs diagonally


where n = u - 4 log( 1 + 4 m ) . Furthermore, i f H A depends analytically on parameters, then so does GA(z). In norms defined in (2.17), for any 0 < y 5 0' there is a constant CO such that

(5.3)

Proof This proof is neatest via path expansion of the resolvent ( H A - z l ) - ] . First subtract the diagonal from DW,

H A = V(0) + DW = V1 + DWI

where V I = V + diag DW. For E 5 (1/4)S the elements of the diagonal matrix IVl(x,x)l 2 4 6, while diag D W I = 0, and otherwise DWl(x,y) 5 E ~ - ~ I ' - ~ ~ . Certainly IVi'(x,x)I 5 4/(36). Consider the Neumann series for the Green's function GA(z)(x, y ) = (HA - zl)-I(x, y ) = (VI - 21 + DWl)-'(x, y ) , considering DW1 as the perturbation,

3

(HA - = (vl - 21 f DW1)-' = (vl - z1)-'(1 + DwI(vI - Z l ) - ' ) - ' .

Set T = -DWI(VI - formally,

which also vanishes on the diagonal, so that at least

(5.4) n=o

We of course have IT(x, y)I 5 4 ~ 6 - ' e - ~ I ~ - Y l . We sum (5.4) over paths, preserving an exponential decay estimate for the Green's function. Consider a path P = (x, P I , 02,. . . y ) from x to y in the domain A, with the number of steps taken #P = n, and with total length [ ( P ) = ID; - P j - l I . Then

(5 .5 ) m x , y ) = c n U P ; - 1 , P;) 3

pathspx-y j = l #p=n

where we make the identification PO = x and Prr = y . Inserting the estimate on 1 T(P,- I , P;) 1 from above we find,

paths PIX- Y # 8 = n


The full Neumann series is now considered.

(5.7)

This gives an estimate of the Green's function in terms of the number of paths from x to y of length L with n steps. For our two-dimensional lattice a sufficiently good upper bound on the number of paths from x to y is #{p : e(p) = L,#p =

n ) 5 4" (:I;)'. The resulting estimate of (5.7) is

OO

5 S(x, y ) + c ( 4 2 ~ S - ' ) ( 1 + 4 f i ) 2 ( L - 1 ) e - 0 L . L=Ix-yl

This sum converges if < 6 giving the estimate

Combining this estimate with (5.4) yields (5.2). Estimate (5.3) then follows from (5.2) by way of Proposition 3.1. These estimates are uniform in parameters, as long as the hypotheses of the theorem remain valid, therefore analyticity properties of H A are preserved in GA(z).

COROLLARY 5.2. In fact one sees from the proof that a stronger estimate is true

Let A be any nonsingular domain and let EO C &, U A . Consider the operator HE^ = V W ) + DW(u0). If (r i + p 0 ) G 5 &/2, then for every 0 E D(No,po),


IV(Q)(x,x) l > do/2, for every point x E &. This follows from the form of V(Q) , and the fact that lV(wl)(x,x)I > do by the (do,L&nonresonance hypothesis. If

all sites of Eo are nonsingular. The operator DW(u0) is bounded from Xu+yo to ZU provided o < o* -6' as in Lemma 3.3. Indeed. if I luoll/yo < 1, hypothesis H2 implies that

I(Dw(u~))(x,Y)I 5 ( ( ~ w I I p I I ) / y ~ ) e - ~ n ' ~ - ~ ' .

That JJuol//yo < 1 follows from the bounds on uo in Section 3, and the hypotheses of Proposition 4.6, if we note that yo > i?/128. In order to make the labeling of the indices consistent, define 0-1 = 00 + 6y-1, with y - I = oo/64. Both of these definitions are consistent with the original inductive definitions in Section 3.3. Thus, we can apply Theorem 5.1, set z = 0, and obtain the following estimate.

PROPOSITION 5.3. Let A be any nonresonant region, assume {w,} i s a (do, LQ)- nonresonant sequence of frequencies, and let Bo be the region dejined in Section 2. I f in addition

(5.10)

and i fEo C & U A then

Remark. If the nonlinearity in (2.9) has no quadratic term, the estimate on DW(u0) can be improved to

This allows us to replace hypothesis (5.10) by

(5.12)

and still obtain the same estimate on the Green's function.

1464 W. CRAIG AND C . E. WAYNE

Remark. This proves (3.21) on our initial domain, if So 5 d, and yo 2 ro. The first of these definitions follows from the definitions of SO and d, provided a > 7, while the second follows from the observation in Section 2 that YO = &(2-(zc’), while yo is independent of Lo. Thus for Lo large, this inequality will be satisfied.

Note that the hypothesis is independent of the domains Eo. This emphasizes the fact that it is small values on the diagonal V(R), and the associated singular sites which play a major role in determining the decay of the Green’s functions for HB,(P , 0, un).

At each level of the induction there are two steps to perform. The domain under consideration is extended from B, to B,+1 G Z+ XZ, and the Green’s function G,+, on the larger domain must be constructed and estimated. Secondly, a Newton step is performed to construct the next correction v,(x, p, R) to the approximate solution u,(x; p, R). In the first of these steps we construct GB,+I(P, R, u,; z ) from G x for (p,R) E N,+1. This is the most detailed construction as there may be many resonant sites in the region B,+I\B,. The second step also requires us to estimate the inverse of the linearized operator H K ( ( P , R; u), for u = u,, + v,. Due to the rapid convergence of Newton’s method the adjustment v,(x; p, 52) is very small and once one has control of GB.,I(P, Q, u,; z ) the construction of G z ( p , 52, u,, + v,,; z ) is relatively painless.

5.2. Domain Extension

The induction hypotheses produce a set of parameters (p,R) E N , such that U A, where A is a nonsingular region, and for any IzI < S,/2 the for any E,, C

Green’s function has the estimate

(5.13)

Take Lo large enough that 60 < d,/Co. Proposition 5.3 implies that this estimate also holds for n = 0. In this subsection we extend this induction estimate to the domains E,+1 C B,+1 U A, with tolerance Sn+l as long as A remains nonsingular, and all sites within B,+l\B,, = An+l obey the induction hypotheses (j.1) and (j.2), j = 1,. . . , n. In this process we see that the geometry of the resonant sites is important, as well as the severity of the resonance.

Let { S k } be the set of singular regions of HB,+, (P,R,u , ) within A,,+,, and define Ce,,,,(Sk) to be balls of radius (,,+I about sk, isolating neighborhoods of the singular regions from the rest of By assumption (n.2), for all parameter values (p, 0) E N,+1 these regions are well separated, and C(Sk) are disjoint. The regions Sk, of course, vary with 52. The inductive estimate on the Green’s function will be proved by decomposing the domain B,+1 into subdomains, constructing Green’s functions on the subdomains, and then patching the full Green’s function together using a resolvent expansion. Related estimates and resolvent expansions

-


have appeared in work of Frolich and Spencer: see [9]. First we consider the Green's function on the neighborhoods Ctn+ I (Sk ).

LEMMA 5.4. Suppose that the parameters satisfy the hypotheses of Propo- sition 5.3. Let S C A,+I be a singular region, with dist (spec (Hc,~+,),O) 2 & + I

and (p, 0) E N,+ I . Assume that (j.1) and (j.2), j = 1,2,. . . , n hold. Then there exists a constant C > 0 such that for IzI < ;&,+I, and

( 1 )

(5.14)

then

(5.15)

This implies

Remark. In the case where the nonlinearity in (2.9) contains no quadratic term, condition (5.14) can be replaced by

2 (5.16) r; < dsYo

Proof The proof uses resolvent identities to decompose the Green's function into several parts, allowing matrix elements to be estimated. Let S be the singular region itself, and D = CtnLl (S) - S the annulus surrounding this region. Induction hypothesis (n.2) implies that the singular regions are so far apart that the annulus D contains no other singular sites. The linear operator HcI,+,(s) is decomposed into block diagonal and off diagonal parts,

The operators HD and Hs are the restrictions of Hci,+,(s) to the subdomains D and S, and the term TDs accounts for the interactions between these domains. There is accordingly a decomposition of the resolvent, giving the first resolvent identity,

(5.18) Gc,,~+,(s) = GD @ Gs + GD @ G s ~ D ~ G c ( s ) .

(Again, to save space, we write G,-(s) rather than Gc,,,_,(s).) Iterating this and its adjoint we obtain the second resolvent identity

I466 W. CRAIG AND C. E. WAYNE

The Hamiltonian H D has diagonal elements which are nonsingular and satisfies the hypotheses of Corollary 5.2. Thus

The set S consists of only one or two sites. The interaction term has zero diagonal, and decays exponentially off diagonal;

We shall treat estimate (5.15) in three cases. If both x,y E S , then lGctn+,(~)(x,y;z)I 5 2/Sn+l, since by hypothesis

dist(spec(~C,"+,(s)),O) > Sn+l The estimate (5.15) follows since Ix - yI 5 1. Sec- ondly, if x E D and y E S (or vice versa), the first resolvent identity is used

(5.20)

Note that Go @ Gs(x, y ) = 0 in this case and we have used the fact that 1 q - y I 5 1. Finally, consider the case when both x , y E D. Using the second resolvent identity,

(5.21)


We have used the fact that TDs(p,q) = 0 if both of its arguments are in D to eliminate the term linear in from (5.21). We estimate this exponential sum with a loss of decay yn-l . This results in a bound on (5.21) of the form

Thus inequality (5.15) follows from (5.20) and (5.21) using J(p(( < ro < d,yi .

In fact a useful corollary of the proof describes in more detail the structure of the local Green's function in neighborhoods containing only one singular region.

COROLLARY 5.5 . Assume that the hypotheses of Lemma 5.4 hold. Suppose that S 5 A,+] is a singular region such that Ccn+,(S)\S = D is nonsingulat Then

(5.24)

Continuing the proof of the induction step, the neighborhoods Cu,,,, (S,) are patched back into the region & + I . This process is performed using a Neumann series based on the resolvent identities. In fact this expansion has superior convergence properties if two sets of domains are used in alternation: the domains

- C,+, (Sk) , and Ctn+,(Sk), and the same sort of domains made from Bn+l by excising only the resonant regions {Sk}, rather than the disks C(Sk). Of course the Green's functions G{s,} also satisfy (5 .13 , by virtue of induction hypothesis (n.2) (ii).

THEOREM 5.6. Suppose that the induction hypotheses (j.1) and (j.2), j =

1,. . . , n, are satisjed and the hypotheses of Proposition 5.3 hold. Let A be any additional nonresonant domain. There exist constants C and CG independent of n such that if

10 rOCe-Yn-I~ll+1/2 7 < 1 , 15 s1, dsYo 6 n + 1 ~ n - I

W. CRAIG AND C. E. WAYNE 1468

and rOCC;e-Yn-l en+ 12

2 34 5 1 , Sn+ 1 Y n - 1

and if En+ 1 C U A, then the Green's function GE"+~ (x, y ; z ) satisjes

(5.25)

We shall prove (5.25) by estimating I GE"+~ (x, y ) 1, and then applying Proposition 3.1.

A central tool in the proof of this theorem is an estimate of the many exponential sums that arise in resolvent expansions. In particular, there arise many sums over classes of paths which, by the arrangement of the decomposition of domains, are forced to contain a "long step." In the induction the size of this step will typically be en+1/2. The following elementary lemma concerns such an estimate.

LEMMA 5.7. Let {P}",e be the collection of all paths from x to y in a domain B C Z+ x Z, with aJixed number of steps, #P = n, and a length e(p) of at least C . Then for any 0 < y < n,

Proof

by an elementary exponential estimate, and the lemma follows.

Proof of Theorem 5.6: We estimate GE,,+~, for an arbitrary subset E,+I C U A by a path expansion. The paths appearing in the expansion of GE.+] are

a subset of those appearing in the expansion of GB,,I"~, so the estimate on GE,+~ follows from that on GG"~. The procedure is to isolate each singular region Sk C A,+I with neighborhoods Cen+,(Sk). Since the radius of Ce,+,(Sk) is en+, the resulting alternating expansions will be composed of elements whose path


expansions consist of long paths of length O(tn+I). The result is a convergent Neumann expansion and good estimates of exponential decay.

The resolvent expansion for Gs,,,uA(~, y; 2 ) is the Neumann series constructed from resolvent identities where two sets of domains are used alternately. Let B = E U, D, = E U, D,, where E , {D,} and E , {D,} are collections of disjoint domains of the lattice. Then we can express

(5.26) HB = H E $ j HD, + r E D = HE $ j HE, + r m . Applying the first resolvent identity to both of these relations

GB = GE $1 GD, + GE $1 G D , ~ E D G B (5.27)

Alternately using the first and second of these formulas, a formal expansion of the Green's function is obtained.

= G, @ @j, f G, @ , Go, rmGB .

GB = GE @ J GD, + GE $1 G D , ~ E D G F $1 G q X

(5.28) f C ( G E @ J GD, rEDGF @ J G,, rm)'

X (GE GD, + GE G D , ~ E D G F $1 G q ) Y = 1

We shall use this expansion with the collections {D,} and {q} the set of singular regions, s k , or their neighborhoods Centl (Sk).

For situations in which all operators in this expression have exponential off diagonal decay, and where there is a large distance between domains E and D,, the use of two decompositions of the domain in formula (5.28) results in an expansion with superior convergence properties. The proof of the estimate of GB,,I,,* consists in making choices of the sets E , {D,} and E , {D,}, and estimating the formal sum (5.28). Cases are chosen as follows:

Case 1. If dist(y,Sk) > ;e,,+l, for all singular regions sk c A,+I , we set D, = S,, where the index runs over all singular regions in A,+I. Then define E = En+) U A - (U,D,). Also let D, = Ctn+,(SJ), and

Case 2. If dist(y,Sk) 5 ?e,,+l, for some singular region Sk C A,+I, chose D, = Cend,(SJ), for each of the singular regions S, C A,+I, and E = B,+1 UA-(U,D,). Similarly, taken, = S, , andE = B,+1 UA-(U,D,).

Address first the block diagonal term GE @, Go, of (5.28). In both cases, the domain E c U A, for some set A consisting entirely of nonsingular sites. Thus, by the Theorem 3.8, I / GE ll,.,-,-2yn 2CE/(S,y!!l). On the other hand, GD, is either of the form Gs,(x,y;z) or Gc( , , , (~o(x ,y ;~) , so Lemma 5.4 implies that )IGD, llgn-,-4y,, , 5 C / ( b , + ~ y , - ~ ) for all singular regions of A,,+1. Thus, we have proved

_ _

E = B,l+l U A - (U,D,). 1

12


LEMMA 5.8. Ifthe hypotheses of Lemma 5.4 hold then there exists a constant C > 0 such that

(5.29)

The term in (5.28) linear in the coupling r involves the choices of decomposition more directly. In Case 1, y E U E. Assume that it is in E. Since GE has block diagonal form, q E E, implying that q E E . Since rED(p,q) = 0 unless p E D, we have p E D, for some j . Thus in this case we have

(5.30)

from Lemma 5.4. Since dist (y ,Se) > tn+1/2 and p E S , for some j , then ly - q1 + 1q - pI > tn+1/2. We refer to this as the long step in the sum. The geometry of this path is illustrated in Figure 1.

This is just the situation addressed by Lemma 5.7, thus the sum (5.30) is bounded by

Figure 1. A typical term in the expansion with p E E and q E D.


(5.31)

If y E D instead of y E E , then q E D and we consider two possibilities, q E D U D and q E U E . If q E E , then p E D (since rED(p, q ) = 0 otherwise), and we would obtain a long step exactly as above. If the second possibility holds, then q E S j for some j , and hence Iy - q ( > en+,/2, so that this is the long step. In both situations the sums in (5.30) are estimated as in (5.31).

In Case 2, we again have y E E U D. Assume that y E 0. Then q E D, so q E D, and thus p E E . We see that Iy - q1 + 1q - pi > tn+1/2, once again giving a long step, and we can bound the exponential sum as follows.

(5.32)

Figure 2. A typical term in the expansion with p E D and q E E


If y E E, we again have a long step, and the estimate (5.32) again applies - the details are left to the reader. Combining the estimates (5.31) and (5.32), we have the following decay result.

LEMMA 5.9. constant C > 0.

(5.33)

then

(5.34) I(GE

We used no

If the hypotheses of Proposition 5.3 hold, then there exists a independent of n such that if

roc e-Yn- len+1/2 < 1 , < 1 and 15 &Yo 6 n + 1 Yn- I

information about the placement of x in this result, thus (5.34) holds uniformly in x.

Remark. In the case where the nonlinearity contains no quadratic terms, the second inequality in (5.33) can be replaced by

(5.35)

Finally, we bound the infinite series in (5.28). The estimates are much like those immediately preceding. We shall prove

LEMMA 5.10. Assume that the hypotheses of Lemma 5.9 hold, and that the regions D, E , D and E are given as in either Case 1 or Case 2. There exists a constant C > 0 such that

I GE @ j GO, rEDGE @ j Gzj,rE(w, Z) 1 (5.36)

Proof Consider first the situation where D,E,D and E are in Case 1. In Subcase 1, suppose that z E 0. Then,

I GE @ j GD, rEDGE @ j GD, Z ) 1 (5.37)


Here the restrictions on the summation arise becawe of the block diagonal form of the Green's functions in the product, and the fact that r E D and Tm vanish unless one of their indices is in E (respectively, E ) , and the other is in D (respectively 0). The paths in the summation have a long step between r and p . Thus the bounds derived above imply

(5.38)

Lemma 5.7 is used above in estimating the exponential sums over paths with large steps. In Subcase 2, z E E, and we have to break the expression for GE @, G D , r E D

G, @, G,,T=(w,z) into two sums, one in which r E 0 , q E D n E , and p E D. and a second in which r E D,q E D fl D, and p E E. In the first there is a long step from z to p , and in the second, from z to q. In both cases, using Lemma 5.7, these sums can be bounded by the right-hand side of (5.38).

If the placement of x, v is in Case 2 , a similar computation, using the estimates of (5.32) leads to an identical conclusion - we omit the details.

Remark. If the nonlinearity in (2.9) contains no quadratic term, the left-hand side of (5.36) may be replaced by

(5.39)

The full resolvent expansion (5.28) is now easily estimated. The Neumann series for the full Green's function is

(5.40)

Using Lemma 5.9 to estimate the factors linear in r E D , Lemma 5.10 and the operator norm estimate Proposition 3.1 to estimate the factors ((GE @, G D , ) r E D


(GE 8j GD,)T,)e, and Lemma 5.7 to bound the sum over z , this sum is bounded by

The hypothesis of Theorem 5.6 guarantees that this sum will converge, and, when combined with Proposition 3.1, this suffices to prove Theorem 5.6.

5.3. The Newton Step

The second step in the induction is to perform a correction to the approximation given by Newton's method. That is, the new approximate solution is u,+1 = u, + v,, where v,, is obtained as a solution of a linear equation on the domain G, using the Green's function GB.+I(P, R, u,). Linearizing the nonlinear equation about this new approximation changes it; H~n+l(p, 0, un+l ) and its inverse must be estimated. Due to the rapid convergence of the Newton scheme these corrections are small and the estimates are not difficult. From Section 3, we know that

(5.41)

The following estimates are designed to correct the operator H B , + ~ (Un+l) and the Green's function GB.+I due to these adjustments - in fact they are valid for any sequence v(x) satisfying inequality (5.41).

LEMMA 5.1 1 . Let A be any lattice region. If one adds to the function u,(x) any v(x) satisfying the estimate (5.41), the mod$ed Hamiltonian has the form

(5.42) H E , + I ( U n + V) = H E , + I ( ~ I ) -I- Rn+l(Un,V) 9

for any En+l C B,+1 U A. Then R,+1 satisfies the estimate

(5.43)

Proof The linear operator HB(u) = VB(R) + DW&, u) consists of a linear diagonal part, which is independent of u, fv,, and an off diagonal piece DW(cp(p)+ u). From estimate H2, DiW has operator norm bounded by CW llpll y e 3 on The estimate on R,+I follows from Taylor's theorem.

When (p,R) E Nn+l the spectrum of is bounded by 6,+1/2 away from zero. The induction provides for perturbations R,+ 1 satisfying (5.43), which are substantially smaller than this. The Green's function G~.+,(u,+l) can be constructed from GE,+~(U,) and R,+1 via Neumann series for z in the set { z ; IzI 5 6,+1/4}. This gives the following result for the construction.


THEOREM 5.12. Under perturbations of size (5.43), the Green's function remains a bounded operator on 2Z'u,-2y, for all IzI 5 b,+1/4 , as long as

and one has the estimate that

(5.44) 2cI;"+" I1 G E . + , ( ~ , + l ) IIuT.-2y,5 ~

&+lYA2 .

Proof A simple Neumann series gives that

GEn+I(U, + V ) = GE,,I(U,) ( 1 + &+iGc,+l(Un))-i,

so that it suffices to bound 11 R,+IGE,,,(u,) l \ u n - 2 y n 5 1/2. The stated condition on the constants assures that this will be the case.

Note that Theorem 5.6 and Theorem 5.12 imply Theorem 3.8 which we used

The same method allows us to carry through an estimate on the Green's func- repeatedly in Section 4.

tion on nonsingular domains. One has the

COROLLARY 5.13. Let A C Z' x Z be a nonsingular domain and u, =

ug + cgri v,, for n z 1. ~f

and if ( z I < 6,+1/2, then

The proof is again a simple application of Neumann series - we do not repeat it.

6. Final Reckoning

The proof of convergence of the induction is based on a proper choice of the constants that appear in the analysis throughout the paper. There are roughly three requirements to be satisfied. First, the inductive constants a,P, and K must


be chosen so that the Newton iteration method is convergent. This involves estimates of the truncation in approximating the nonlinearity, the excision process in the parameter domain, and the iterative construction of the Green’s function. Sec- ondly, the initial approximate bifurcation problem and parameter neighborhood must be sufficiently large so that, even after the excisions of the induction steps, the remaining solution set has large measure. Finally, we must be able to carry out the analysis of convergence for an open dense set of nonlinearities g(x, u) for the wave equation, thus requirements of Proposition 2.4 and Theorems 2.8 and 2.9 must hold. In this subsection we show that all the requirements of the paper can be simultaneously satisfied. We accordingly make choices of the inductive parameters a,p, and K , and show that the principal requirements on the parameters ro ,po and do and d, reduce to a condition that b, the initial radius of an approximating lattice domain, be sufficiently large.

Before the induction starts, a bifurcation analysis is performed on the initial domain Bo of radius Lo. The a priori estimates on the initial domain require that the induction starts with uo E Xu,, no + yo < n, - 1/b. Setting do = for some 1/2 < 17 < 1, and ro = po = cL,, , for c some small positive number, we are able to satisfy the hypotheses of Lemma 3.3 and Lemma 3.5,

-a+?)

b&iZ < cdo (6.1)

&ri + p i ) < cdopo . These definitions of ro and p o determine the maximum and minimum values of R (Omin and Om=). Several of the estimates in Section 4 (e.g., (4.19)) depend on these values. We choose ro and po sufficiently small (or equivalently, b sufficiently large) that w1/2 5 R ~ n < Om,, 5 2wl. Thus, all dependence on flmin and Rm, can be replaced by a dependence on w1.

We next turn to the choice of the inductive constants and the question of convergence. The sequence ug E t2 (Bo) f l Xuo is an exact solution of the restricted equation &F(uo) = 0, thus there exists co > 0 such that

l l ~ ~ ~ o ~ P ~ ~ ~ ~ l l o o = Il(1 - no)w(cp(P) + uo)Ilco

as long as LO 2 L , , is chosen sufficiently large so that o* - 1/b - yo 2 00 > 5/2. We have used Lemma 3.2, and H1 to estimate the nonlinear term. Make the choice

E0 = e - c o b ,

We now verify that the hypotheses of the propositions in Section 3 are satisfied. First consider Proposition 3.6. Inserting the definition of the various inductive


constants, and defining D = IaI + p, the total number of differentiations, the hypothesis of this proposition reduces to

X c ( 2 a ( ] + I ) G + a )D(2a( j+ I )~)c:l no ~ 12 (64(j + 2)2)12e0’K’Lo j = O

< c ( D ) ~ - Q ~ / ~ .

Noting that (2a(i+1)~+a)De-CoK’Lo/4 I Cl(D), independent of j and Lo, and that for b sufficiently large.

X

C( 2 4 + 1 )G)CA+ 1 go - 12 (64(j + 2)2)1ze~c0K’Lo/2 < C? , ]=o

since the left-hand side of this expression can be majorized by a geometric series, we see that the hypothesis of Proposition 3.6 is satisfied if L-, is sufficiently large.

Similarly, consider the hypothesis of Proposition 3.7. Upon substituting the definitions of the various inductive constants, it reduces to

Once again, CiY=0(2(i+1)LO)C~+’~013(64(j + 2)2)13e-c0K’Lo/2 < C for sufficiently large, so this estimate reduces to e?°K’h/2 S CL;(, which is clearly satisfied for sufficiently large. The hypothesis of Proposition 3.9 is verified in the same way.

The final estimate to be verified in Section 3 is the hypothesis of Lemma 3.1 1:

-7 1/4 C p o - E o << L(-u ,

4 + 2 ~ - ~ which reduces to e-cOb/4 << C r ,

as the hypotheses of Proposition 4.1 are satisfied;

, and this is obviously true for large L-,. For n > 0 the iteration will have a supergeometric rate of convergence as long

(6.2)

Using the definition of the inductive constants, (6.2) will follow from the pair of inequalities

(6.3)

and

(6.4)


The governing factor of the left-hand side of (6.3) is e-cb2"/1n12, while from the definition of EO that of the right-hand side is e-coLO(l-(l/K))Kn. If necessary decrease co so that CO( 1 - ( 1 / ~ ) ) < 2c. For L , sufficiently large the estimate (6.3) will hold for all n > 0, uniformly in Lo > L , .

A similar discussion verifies that (6.4) will hold for all n > 0 if Lo > L , is sufficiently large. The right-hand side is governed by the factor e (y0b(2 -K)Kn- l )

which is much greater than 1 since K < 2, while the left-hand exponent grows at most linearly in n and logarithmically in Lo. For > L , , L, chosen sufficiently large, it too will hold for all n > 0.

Note that (4.2) and (4.3) have already been verified in our discussion of the hypotheses of Section 3.

In order to make the excision process work, relations (4.7H4.10) of Proposi- tion 4.6 will have to hold. Set

2d, = do (= Lo') . If we choose rp small enough that 1 -rp > 7, the first condition in (4.7) is satisfied if Lo is sufficiently large. Since the parameter d is at our disposal, we satisfy the second condition of (4.7) with the definition d = C27(fl+1)L7p-1 N o . The first condition of the second line is satisfied for some CO < 1 - p/ log(2). Substituting the definitions of do and ro into the second condition there, it reduces to the inequality (CO log(2) + p)(1 + 7) < 4 + 7, which can be satisfied by picking co and p small. The following three relations follow immediately from the definitions if Lo is sufficiently large. The final relationship in (4.71, C(l + (h/dO))Pn+l < 6n+1 reduces to Lop0 < cLido upon substituting the various inductive constants into the inquality, and this follows from the first inequality in (6.1).

Without yet discussing the measure of the remaining parameter region (i.e., (4.8)-(4.10)), we verify the rest of the requirements of Section 4. In order to separate the localizing neighborhoods Ce, ( S ) , the hypotheses of Lemma 4.8 must be satisfied. They follow, however, from the estimates of (4.7) which we verified in the previous paragraph. Note that the hypotheses of all the remaining lemmas in Section 4 follow from those of Proposition 4.6.

In order to construct the Green's function using the procedure of Section 5, we ask that h > L , which is sufficiently large so that Cwro/(Z2d,) 5 Ca, . This permits the construction of the Green's function on nonsingular domains. If we choose

uo + 57-1 = (32) 00 < u, - 210g 1 + 4 Cwro/(z2ds) ,

then the restrictions on DO in both Theorem 5.1 and Proposition 5.3 can be satisfied. The requirements of Proposition 5.3 and Lemma 5.4 reduce to ro < Cdo and

ro < -yi d, if we insert the definitions of the inductive constants. These inequalities are both satisfied as in (6.1) by an increase in L , if necessary. The more intricate patching techniques of Theorem 5.6 require additionally that

( 7) 35


As before, for any choices of a,P there is an L , such that this will hold for all n 2 1, uniformly in > L , . In addition, in Theorem 5.6 we required that

ro,-C;e - Y.- I en+ I 12

2 34 51, 6n+ 1 Y n - 1

but this follows without problem from the previous inequality. Using (6.5), the hypotheses of Lemma 5.8 and Lemma 5.9 follow accordingly.

The second type of extension of the Green’s function is addressed in Theorem 5.12, where it is asked that

(6.6)

Using the definitions we see that this follows from

which can be uniformly satisfied for > L , , for all n h 0 as long as L , is sufficiently large. Corollary 5.13 follows from Theorem 5.12 if the correction term v j is small in norm. That is, given (6.2) and (6.3), we have

which is even easier to satisfy than (6.6). The remaining requirements that are placed on the inductive constants come

from Proposition 4.6, where an estimate is made of the measure of the parameter set after the n-th excision. This will involve the choice of a and 7 . To satisfy (4.8),

using the geometric growth of e j . Substituting the definitions of d and N o , this is bounded by


Since ro = c,$~+'), the above expression is bounded by i-i if p < colog(2)/(3 + 277) and d b is sufficiently large. We constructed the Nfl+l by first constructing two intermediate sets JV,,+~ and NFil. The estimate of (6.7) bounds the measure of the excisions that are made in constructing Nrj To control the size of excisions in constructing Nn+ we require

(1)

(2)

As long as we satisfy the requirements that

a + 1 - 4p - v > 2(2 + 17) a + 4 + 217 - 4p - v > 2(2 + 77)

then (6.8) will hold uniformly in

X of the surface is restricted from being too small. Using the above choices,

> L, and n > 0. Addressing (4.9), we see that the inequalities will be satisfied if the curvature

This will hold for all n B 0, uniformly in > L, if the curvature satisfies

for some 0 5 v < 1-77. The exponent v appears in the twist condition of Theorem 2.8, for a bifurcation surface must intersect the sets Nn+l = N!il n NFj1 to exhibit a solution of the equation. To estimate the measure of this intersection, we show that for a and T reasonably large, (4.10) is satisfied for some exponent p. The first inequality is

using the geometric growth of ti. Using the definition of d and the lower bound on 1x1, this expression is bounded by $+'), provided that v - (1 + colog(2)/B) < -2(1')(2 + 17) and 2(1+ p)(2 + 17) < 4 + 217 +(1- p)(1+ co log(2)/p). Clearly by


sufficiently small choice of p, these inequalities can be satisfied. Considerations of the second inequality are similar.

(6.1 1)

1 . ~ J' ( ~ g ( n + 1 - 4 @ ) / 2 -(a+2(2+q)-44)/2 +Lo 17Ll

where we must require that cr > 4p. Allowing 1X > Lo-", this expression will be dominated by ro ( I + r ) for

cr - (v + 277 + 4p + 3) 2 0 + 77)

O < p <

The bifurcation surface has the form R(p)wl +Xllp112(1 +Y(p)), where 1 . 4 0 1 ~ 2

S 1/2 as long as 0 5 llpll < rdfE), E = ( 1 + 77 + v)/(4 + 2v). From (n.2) (iii) this results in a solution surface with positive measure intersection with the set JV, as long as E < p. It is the radius $ + E ) = r* that appears in the statements of the theorems of Section 2. Clearly a choice of cr and T large, and p small is available so that these inequalities will hold throughout the induction.

The last relations that should be verified pertain to the the density of the nonlinearities to which the results apply. We require that do = o(&' /~) in order to satisfy the hypotheses of Proposition 2.4. This gives a genericity result for the coefficients gl. This is the origin of the restriction 1/2 < 77 < 1 . Secondly, we ask for bounds on the curvature in the twist condition. Setting 1x1 2 &" with v > 0 gives a bound which is decreasing in Lo, and again there is an open dense set of nonlinearities possessing a sufficiently large twist for Theorem 2.7 to apply.

7. Analysis of the Nonlinear Operator

For the nonlinear wave equation (2.1) it is natural to assume that the nonlinear term g(x,s) is periodic in x with period T , and analytic in the region {(x,s) E C2;JIm X I 5 a}. These conditions translate into conditions for the nonlinear operator W applied to sequences defined over the lattice 27' X Z. In proving these estimates, we fix an index c L < 5, and then give estimates that will hold for all o < o * . The constants will depend upon 5 and c*, but will be uniform in c < D * .

This section also addresses several of the properties of the nonlinear wave equation that have been stated in Section 2, including the properties of genericity of (do, Lo)-nonresonant frequency sequences {w,>,"=~. This will finish the


proofs of the main theorems, Theorem 2.1 and Theorem 2.2. Finally, there is a proof of Theorem 2.11 on the accumulation of periodic orbits at zero in the fully nonresonant case.

7.1. Verification of Hl-H3

We consider two cases of boundary conditions for equation (2.1) for which hypotheses Hl-H3 are satisfied by the associated nonlinear operator on the lattice. The first case is for periodic boundary conditions on the interval [O,n], and the second is Dirichlet boundary conditions imposed on the endpoints 0 , ~ . In this second case there is an additional restriction on the non-linearity.

Periodic Boundary Conditions

Assign periodic boundary conditions to the Sturm-Liouville operator L(g1) =

-2 + g l ( x ) on [O,n]. The potential gl is assumed to be periodic, of period T , and analytic in the strip {x I IIm X I < 53. The spectral asymptotics for Sturm-Liouville operators guarantees that there are at most finitely many negative eigenvalues. Label the eigenvalues

I, ... .

and let {I,!J,(x)},"=o be the corresponding sequence of eigenfunctions. The main technical result of this section is that these eigenfunctions have uniform properties of analyticity in {x I IIm X I < a}.

PROPOSITION 7.1. Expand +,(x) = Ern $,(m)eimx. Then for a* < a, there are constants Cl(gl ,a , ) and C2(gl ,a , ) , (independenr of n) such that for m > 0 one has

Remark. The eigenfunctions I,!Jn(x) for the periodic problem are analytic in the strip {x I IIm X I < 5) - thus this result "almost" follows from analyticity of the eigenfunctions. The one thing that must be checked is that the constants Cl ,*(g l ,a) can be chosen uniformly in n. The proof of this proposition will be given below, following a discussion of some of its consequences.

From this proposition we obtain the following useful corollary. Recall that we defined the complex domain 9o = {(x, 6 ) E C2 1 IIm X I < a, IIm <I < a} . If f(x,<) is analytic on gU, and periodic with period n in x and 2n in <,


define \ \ f l lo,x = SUP(,,^)^^/, l f ( x , ()I. Let f be the coefficients in the eigenfunction expansion of ,f.

COROLLARY 7.2. There exist constants C l (g l ,a , - a ) and Cz (g l ,a , - a ) such that for all 0 < y 5 a < a*, l l f l lu -y ,x < ( C l / ~ ) I I f l l ~ and I l f l l u - y < (C? /Y) I l i I I 0-y.x .

Proof One just writes out f (x ,<) in terms of its eigenfunction expansion, substitutes for the factors I+!I~(~) their expansion in terms of exponentials, and then uses the estimates of Proposition 7.1 to bound the resulting sums. To estimate f one writes down the definition of this quantity, and again bounds it using the estimates of Proposition 7.1.

There is a second useful consequence of Proposition 7.1.

COROLLARY 7.3. There exists a constantC3(g,,a* -a ) such thatfor a < u * , The eigenfunctions { i j~ , , }& satisfy (2.19) with C3 = C j ( g l , a , - a).

Proof: Once again, the proof is immediate if we expand the eigenfunctions rL,, in terms of exponentials and then use the estimates of Proposition 7.1.

Remark. As a consequence of this corollary, if we take C3 = C3(gl, a I - CT),

then the I I i f 1 1 I c ? . ~ norm dominates the Ilfllu,x norm. With the proposition in hand, verifying the hypotheses on W is straightforward.

Assume that u E Xo with IIuIIu 5 rg and I I I U I I I C ~ . ~ S ro, and that g(.w,u) is analytic for (x.u) E { ( x , u ) 1 IIm X I < 5 , IuI < l}, for some 0 > 0. Then w(cp(p ) + u ) = ( g - gl )(x. $ ( p ) + ii). (Throughout this subsection, m, Cp, and ii will refer to functions whose lattice representations are W , cp. and u.) From the estimate on I 1 JuI I I c ~ , ~ 5 ro, and the remark following Corollary 7.3, we conclude that the function Cp(p) + ii satisfies

llCp(p) + ~ l l g , x I 2e"(llpll + Y O ) .

Then (g -gl )(x, $(p)(x, () + ii(x, I)) is analytic for (x, <) E 9 g - y / 2 , provided llpll < ro and 4e"rg < 1, which follow from the assumptions of the previous section. Furthermore,

This verifies H1.


Remark. In the case when the nonlinearity contains no quadratic term in its Taylor series, one can improve this estimate to

with similar improvements reflected in the estimates of the first and second derivatives of W.

We now prove that the derivative of W satisfies H2. Let ( j , k ) and ( j ’ , k’) be two points in Z+ X Z. Then

- If we write G j ( x ) = En $j(n)e-im, and a similar expression for t,!~y(x), we have

Since D,g(+(x, E ) + ii(x, E ) ) is analytic for (x, E ) E 9u-y /2 , the double integral is bounded in absolute value by

Thus by Proposition 7.1

with constant C = C(a, - a). Then Proposition 3.1 implies


This, when combined with the previous estimates of IIiiI(o-y/~,x, completes the verification of H2.

Finally we turn to the verification of H3. The boundedness of the second derivative DiW is almost the same as the estimate for H2. so we will not include the details here. The only change comes from the fact that we must use two different norms on the right-hand side of the estimate. This arises because in bounding the exponential sums which enter the definition of our operator norm, we are forced to sacrifice a small amount of the decay of one of the two functions on which the second derivative acts. The decomposition DiW = C + A for the nonlinear wave equation arises from the Taylor expansion of the nonlinear term.

D f ; g ( ~ . (6 + ii) = 2 g ~ ( ~ ) + 6g3(~)(@ + ii) + . . = C + A .

Let : = ( j , k). ;' = ( j ' , k ) with ( j - j ' ) = 0, f 1 and P I + ip2 = refH. The behavior of the bilinear operator C claimed in (2.24) is verified by the integral

(b(z)~C[cp(p), 6(Z')l)O =

(In fact, in the case of the wave equation, this integral vanishes for any j , j ' . ) All contributions to the second piece of the bilinear form A are at least linear in q(p) + u. By hypothesis u E X,, so that a procedure similar to that used for H2, with the use of the estimates in Corollary 7.2 gives the statement (2.23).

Combining these estimates, we have shown that the nonlinear wave equation with periodic boundary conditions gives rise to a nonlinear lattice problem which satisfies the hypotheses H1, H2, and H3.

PROPOSITION 7.4. l fwe consider periodic boundary conditions in (2.9), and if the nonlinearity g(x, u ) is 7r periodic in x and analyticfor (x. u ) E {(x, u ) I IIm X I <

. IuI < l}, for some 5 > 0, then for u* < 5 and for all 0 < u < u* the requirements Hl-H3 are satisfied for the operator W and its derivatives.

Dirichlet Boundary Conditions

The second problem that is considered is for Dirichlet boundary conditions imposed on the interval [O, TI. We assume that the nonlinear term g(x, u ) is odd with respect to reflection through the origin - i.e., g(-x, -u) = -g(x, u). Eigenfunc- tion expansions are thus considered with respect to the Sturm-Liouville operator L(g1) = - + gI (x), with Dirichlet boundary conditions imposed on [0, 7 r ] . Thus in addition to assuming that gl(x) is analytic and periodic in a strip of width 5 about the real axis, it is even with respect to 0 - i.e., g l ( - x ) = gl(x). We label the eigenvalues of L(g l ) in order,

d'

Ld: < Ld: < ld: <, ... .


Denote {+,(X)},"=l the eigenfunctions corresponding to w,. Since gl is even, the eigenfunctions +,(x) can be extended as odd 27r periodic analytic functions, and hence their Fourier series expansions are of the form

The result analogous to Proposition 7.1 is

PROPOSITION 7.5. Expand ~)~ (x ) = Cm,O$,(m)sin(nx). Then for a* < a, there are constants C ~ ( g l , a,) and C2(gl, a*), (independent of n) such that for m > 0 one has

(7.3)

Conversely, if one expands sin(mx) = C, &,(m)+m(x) then

Let f ( x , <) be analytic in go, 27r periodic in x , < and an odd function of x . The analog of Corollary 7.2 is the following;

COROLLARY 7.6. There exist constants C l ( g l , a, ) and C2(gl, a, ) such that for a < a,,

Cl I l f l l o - y , ~ < yllfllo-r 9

and c2

I l f l lo-y < -Il.fllo-y,co * Y

Using this corollary, the verification of Hl-H3 for the nonlinear term W proceeds along lines similar to the proof of Proposition 7.4. We assume that g(x, a) is odd under reflection through the origin so that the nonlinear term preserves the class of 27r periodic functions which are odd in x . That is, g(-x, k( -x , I)) = -g(x , k(x , <)) provided k ( - x , <) = -ii(x, <).

7.2.

We now turn to the proof of the technical results Proposition 7.1 and Propo- sition 7.5. We shall give the details for the case of periodic boundary conditions; of course the case of Dirichlet boundary conditions is similar. Let the function

Proof of Propositions 7.1 and 7.5


gl(x) be periodic with period rr and analytic in the strip {x E @; IIm X I < a}. Consider the Sturm-Liouville problem

(7.5)

with periodic boundary conditions on [O, 71. The eigenfunctions {I),~},& are analytic in the same strip, thus their Fourier coefficients decay exponentially. The point of the proposition is to obtain a uniform bound on this decay. We present here a proof of this fact that is in the spirit of this paper.

Taking inner products of both sides of (7.5) with the exponential functions ek(x) = elkr, gives

(7.6) k2$, (k) + (ek,gI+,) = w%,(k) ,

where $,(k) = (ek, +,). If we now use the fact that {efk'}kEZ is a basis for L2(0,27r), we can rewrite (7.6) as

C ( e k , g l e J ) $ , ( j ) + ( k 2 - w 9 j n ( k ) = o . I

This equation can be rewritten as

(7.7) (V, + D k = 0 ,

where J n E t2(Z) with components $, (k) . V, is a diagonal operator on e2(Z) with matrix elements V , ( k , k ) = ( k 2 - wi), and D is an operator with matrix elements satisfying an estimate

(7.8)

where C1 = Cl(g1). Assume for convenience that n is even. The case of n odd follows from that below with only minor changes in the notation. From the asymptotics of the eigenvalues of (7.5) we know that w', = n. Since k2 - L J ~ = ( k - u,)(k + LJ,), we see that for k f +n, and n sufficiently large, there exists a constant CI = cl(gl) > 0 such that

while IV,( tn , _tn)/nl 5 c1/8. On the other hand, the operator norm of D/n on e2(Z) is bounded by Cl/n, for Cl independent of n. Thus for n sufficiently large so that C l / n < q/8, the operator H , = (V , + D)/n will have a pair of


eigenvalues whose distance from the origin is less than c1/4, and the remainder of the spectrum will be a distance of at least 3c1/4 from the origin. We decompose the lattice Z into two regions relative to these two classes of spectra, S = {n, - n } , and N S = Z - S . The projection

(7.9)

with A the circle of radius c1/2 centered at the origin, will project onto the two- dimensional subspace corresponding to the two “small” eigenvalues of H,; this eigenspace is quite close to e 2 ( S ) . The vector $, E e2(Z) is an element of this subspace, since Hn$, = 0. We estimate &( j ) by

(7.10)

The right-hand side of this inequality is bounded with the aid of the following result.

LEMMA 7.7. Let CT* < 5. There exists No > 0 and a constant C N ~ > 0 such that for n L No,

I ( e j , Pe j ) I 5 cNne-2c*(1 Ij l-nl) .

Define a decomposition of the operator H , relative to the subspaces e2(S ) 8 C2(NS) = e2(Z),

1 n 1 n 1 n

HO = -vn INS

(7.1 1) H s = -Wfl + D) ls

H N S = -W, +D)INs .

The Green’s functions associated with these reduced operators are

(7.12)

The coupling term in this decomposition is defined by the relation

(7.13) H , = H N S 8 H s + I-. Hilbert spaces analogous to those used in Sections 3 and 5 are defined by the norm Ilvllz = CkEz Iv(k)12e2u1kJ, and IIGllu refers to the norm defined in (2.17).


LEMMA 7.8. For < E A, I G j , k ; O l 5 4 / ~ .

Proof This follows immediately from our estimates on the location of the spectrum of H,l.

LEMMA 7.9. and for < E A.

Given 0 < 7 5 CT there exists No = N o ( y ) > 0 such for n 2 No,

lGNs(j,k;<)l 5 -e-(o-?)l;-kl 8

CQ I

Proof This restricted Green’s function is constructed via the Neumann series, with ( H N S - 6) = (Ho - <) + (D/n) . This decomposition gives the series

X

GNS(<) = Go(() E(- l ) ’ ( (D/n)Go(<))J .

For 5 E A and n sufficiently large we shall show that the series converges. The first point is that IlGo(<)l\o 5 4/q , and, using (7.8) and Proposition 3.1, l l D / n l l ~ - ~ 5 I I lgl I I I?/(Tn). Thus for N o chosen sufficiently large, and for n 2 No, we have

;=0

To prove Lemma 7.7 we estimate (7.10), combining these lemmas with the resolvent identity:

(7.14) ~ ( 6 ) = ~ ~ ( 5 ) @ cNS(u + ( G ~ K ) e r G ( g . For k = -+n. the estimate of Proposition 7.1 is simply the requirement of boundedness, so we assume that k f ? n .

Iterating the resolvent expansion, we find that (formally)

G(<) = Gs(<) @ GNS(<) + (Gs(0 @ GNS(<)) r ( G s ( 0 @ G N S ( O ) (7.15) X

Because Gs and GNS vanish unless both of their arguments lie in either the singular or nonsingular region of the lattice respectively, and since r vanishes unless one of its arguments lies in S and the other in N S , the second term in (7.15) vanishes when evaluated at the arguments ( k , k ) , with k E N S . Furthermore, since dist(specHINs,O) > 3c1/4, we have


so the first term in (7.15) also makes no contribution to the projection operator in (7.10).

We now turn to evaluate a typical term in the infinite sum. We use

LEMMA 7.10. If m E S, and 5 E A, then there exists a constant C2 = CZ(g1) > 0 (depending on 5 and r), such that

Proof: Note that

We verify the first inequality - the second follows in like fashion.

( ( G ~ K ) @ G ~ ~ ( s ) ) r) ( P A = ~ ~ ~ ( p , q ; < ) r ( q , m ) . q E N S

By (7.8), Ir(q,m)I 5 (Cl/lnl)e-"lqp"l. From Lemma 7.9 we have (GNs(p,q;Q( 5 (8/cg, )e O-y) lP-q l . Summing over q, and using the fact that m = jln, completes the proof.

-(- -

As a consequence of this lemma we have

COROLLARY 7.1 1. If m E S, C is a non-negative integer and 5 E A, then there exists a constant C3 = C3(gl) >= C2 > 0 (depending on 7f and y), such that

The proof is an easy induction argument, which we omit. Now consider the term in the infinite sum,

{ ( (GsW @ GNS(O) r) (Gs(5) @ G ~ s ( 6 ) ) ) ( k k ) =

G N S ( ~ , j , Or( j , m) ( ( G d O @ GNS(<) ) r) e-2(m, P ) m.p,r€S % / E N S

x Gdp, r; q ) G d q , k ; <) . The various factors in this sum can be bounded by (7.8), Lemma 7.8, Lemma 7.9, and Corollary 7.1 1. Inserting these bounds, we find that the right-hand side of this equality is bounded in magnitude by


If we now sum over q and j , and use the fact that rn, p . and r are all either fn, we find that this sum is bounded by

where we can choose CT * to be any constant smaller than (T - 7. If N o is large, and In1 > No, this quantity can be summed over P 2 2, and the estimate of Lemma 7.7 follows.

The proof of Proposition 7.1 follows from this result. For analytic potential functions gl the eigenfunctions +,(XI are 7~ periodic, and analytic in the strip {x I IIm X I < F } , thus the Fourier coefficients decay exponentially,

IJ,,(rn)l 5 C(n)e-"ln-'mll

The fact that the constant can be chosen independently of n follows from estimate (7.10) and Lemma 7.7. Statement (7.2) and the results of Proposition 7.5 follow from similar arguments.

7.3.

This subsection is concerned with the result Proposition 2.5, and the statements that appear in Theorem 2.1 and Theorem 2.2 concerning the generic nature of the set of nonlinearities to which the existence theorems apply. We claim that this set is open and dense in the topology of uniform convergence in the set of functions d ~ . As we have emphasized in preceding sections, whether or not a given nonlinearity g(x, u) satisfies the conditions of our existence theorem depends only on the 3-jet of g - i.e., on the coefficients gl(x), gZ(x), and g3(x) in the Taylor expansion of g about u = 0. Define the spaces

Genericity of the Set of Allowed Nonlinearities

d: = {g p(x) I ge(x) is analytic for [ImxI < and periodic in x with period 7~ } :

4 = 1,2,3,. . . , with a topology defined by the sup-norm. To prove Proposition 2.5 and Proposition 2.7, it suffices to show that there are

dense open sets 0' C dg, 4? = 1,2,3, for which any function g(x, u) E d z whose 3-jet (gl.gz,g3) E d i x d i x d ; satisfies the conditions of our existence theorem. In the case of Dirichlet boundary conditions the requirement that g(-x, -u) = -g(x, u ) forces gl(x) and &) to be even functions, and g2(x) to be odd. Thus, in discussing genericity for Dirchlet boundary conditions we shall consider the spaces, dz in which in addition to the requirements we placed on di we require the functions ge to be odd (even).

We shall first show that the coefficients gl(x) which give rise to spectra {a,} which are (do,L~)-nonresonant with W I form an open dense set in L'(0,x). We shall then refine these estimates to show that the intersection of this set with d$ is again open and dense.

Y.odd(ev) .


The Dirichlet spectra of the Sturm-Liouville operators

are characterized by the following representation for their spectrum

(7.16) wi = n2 + g1 + d(n ) ,

where g1 = np l s ;g l ( x )dx , and d(n) are terms of a sequence in e2(Z'). By contrast, the set of potentials which are analytic in the strip {x I IIm X I S (TI

are less easily classified by the properties of the sequence {d(n)}. Without yet imposing conditions of analyticity for the potential gl, we shall ask for all spectra which satisfy the (do, Lo)-nonresonance conditions. That is, for ( j , k ) such that l j l + Ikl 5 Lo,

Ik2wY - > do, if ( j , k ) # ( 1 , t l ) and , l kw - j l > do(ljl + Ikl)Y for ( j , k ) + (0,O) .

Define NR = NR(L,, r ] ) to be the set of L2(0, 7 r ) potentials g such that the Dirichlet spectrum {wj>j"=, of g is a (b-',b)-nonresonant frequency sequence for some Lo 2 L,.

PROPOSITION 7.12. For any q > 1/2 and L , > 0, the set N R is open and dense in L2(0, 7r).

Proof For Lo fixed, set do = G', and denote by NR(&) the set of frequency sequences {wj}p which are (b-', ,!&nonresonant. We shall use the fact that there is an isomorphism between the potentials in L2(0, n) with mean value zero, and the sequences { d ( j ) } E e2(Z'); see [18]. The sequences which are excluded from NR(Lo) because they violate the first condition of Definition 2.4 are characterized in terms of their spectral representation (7.16),

(7.17) 2 lk2u; - ~ j l = Ik2wY - ( j 2 + 5) - d( j ) l 5 do ,

for some ( j , k ) , l j l + Ikl 5 b. The condition (7.17) involves only finitely many eigenvalues wj, and clearly leads to a closed condition on both the spaces L2(0, T )

and d ~ . If gl is a potential, corresponding to the sequence { d ( j ) } p l , and satisfying (7.17), the maximal distance (in e2(Z')) to a sequence in NR&) is

The second nonresonance condition, (2.12), is a (finite) diophantine condition on the principal frequency W I . The set of potentials which violate (2.12) is


again closed, thus N R ( h ) is open. Furthermore, if {w,):, is not in NR&) from violating (2.121, then there is an i~’, such that Iw1 - w;I < Cdo and w’, is diophantine. If {iuJ),”=, @ N R ( h ) , first choose w‘, as above. Then note that dist({w;} u {w,>/”=.-, N R ( h ) ) S do#2. Thus, dist({w,}T;,, N R ( L d ) 5 2do4l2.

The set N R = Uli>l/z U b 2 ~ , N R ( h ) is open, and since do = o((Lo)-”*), i t is dense in L’(0, x).

The periodic eigenvalue problem has different spectral asymptotics than the Dirichlet case. In particular the eigenvalues come in pairs clustered about the even indexed Dirichlet eigenvalues. The analog of (7.16) is that

w; = j ? + gT + p ( j )

Ld; = ( j + 1)’ + gT + p ( j ) if j > 0 is even, and

if j is odd. Furthermore we know that c p ( j ) ’ < x, and p(2j - 1) 5 d(2j) 5 p(2j) for j 2 1. The regularity of the potential is more clearly reflected in the sequence {p ( j ) }> , ; in fact, for gl E L’

X c lp(2.j) - P(2j - 1)12 < /+’

and if g I ( x ) E dz, then

(7.18)

for some p . Again consider L , 2 0 and 7 > 1/2, and denote NR,,, the set of L’(0,x)

potentials for which the periodic spectrum gives a (&‘,b)-nonresonant frequency sequence for some 2 L , . Using the Dirichlet sequences { d ( j ) } to approximate the location of the periodic spectrum, and the asymptotics (7.18) to control the deviations of p(2j - l), p(2j) from d(2j), the excision procedure of Proposition 7.12 leads to the following result.

I p(2j) - p ( 2 j - I ) I 5 c,, e-PII’

PROPOSITION 7.13. The set NR,,, is open and dense in L’(0, x).

The next step in proving the genericity results of Section 2 is to show that the set of potentials gl (x) giving rise to (do, Lo)-nonresonant frequencies is dense in dk as well. This implies Proposition 2.5.

PROPOSITION 7.14. The set N R n d:“ is dense in d;“,

Proof: The proof is similar to the proof of Theorem 2.12. Let gl E die’. The corresponding Dirichlet spectrum is

u ; = n ’ + g l + d ( n ) ; n = 1,2 , . . . .


Note that {d(n)} is independent of gl. Thus, if we look at potentials of the form g? = gl + E , their spectrum is

w ~ ( E ) = wt + E = E + n2 + + d(n) ; n = 1,2,.

By excising values of the parameter E , exactly as we excised values of the parameter m in the Klein-Gordon equation, one finds that there exist arbitrarily small values of E for which gl has (do, Lo)-nonresonant spectrum and the proposition follows.

Similarly,

PROPOSITION 7.15. In the case of periodic boundary conditions NR,,, n d& is dense in d&.

Our discussion of genericity is completed by showing that the twist condition is satisfied for a generic set of nonlinearities.

PROPOSITION 7.16. Choose 0 < v < 1 -77 and L , > 0. For eitherperiodic or Dirichlet boundary conditions there exists an open and dense subset of d& X d$ X

d$ (oy, in the case of Dirichlet boundary conditions dp X &$Odd X dp) such that i f the 3-jet (g l ,g2 ,g3) of g is an element of this subset, there exists 2 L , such that the twist condition

holds.

Remark. Proposition 2.7 immediately follows from this result.

Proof From (3.12), JC is clearly a smooth function of g , so the set of nonlinearities satisfying (7.19) is certainly open. If we differentiate (3.12) we find

Denote the set which violates (7.19) by 95’b = {g I IXo(g)I 5 LOv for all Lo 2 L*} . If g E ah, then fixing gl and gz and replacing g3 by g; = g3 + E , with E = O(&-”), we can insure that IJCo(g“)l 2 L;”. Thus the distance from g to the set of nonlinearities satisfying the twist condition is 6‘(&-’). The intersection n h Z L . B b is nowhere dense, which proves the result.


7.4. Proof of Theorem 2.11

Without the hypothesis of full nonresonance it may certainly happen that the point (p, R) = (0, W I ) is eliminated from .it" at some stage of the induction, and the Cantor set of Theorem 2.7 will not have the origin as an accumulation point. In order to prove Theorem 2.8 with the hypothesis of full nonresonance we must inspect the process of excision more carefully. The excisions are of two types: constructing sequentially and JI'":~~!~. The former is simplest to analyze, so we address i t first, Excisions to enforce a diophantine condition on R are taken so that for all ( j , k ) E B,l+~\13,1,

Thus the values of R excised at the n-th step must retain a certain distance from

As long as do/(ljl + Ikl)('+') < c l / ( 2 ( l j l + lkl)(""), for all n 2 No, ( j , k ) E B,,+I\B,~. we have

C1 I R - w , ( > ~

2L:::l" '

and thus (p, W I ) is not excised. This requires the choice 5 5 7 . At the n-th step, yeas{R E ( w ~ - r & i j l + r i ) I do/(ljl + Ikl)"' > IR - ; / k l } 5 doL;'(l + roLn+l). Given a nondegenerate surface V described by R(p) = I,JI + Xllp112(1 + Y(IlplI)), consider the measure of the set B?n+l(rl) = (0 < r < rl 5 ro; llpll =

r , (p. R(p)) E . f , l + l } . If IYLIrf < C~/LF:~'), then no excisions have yet occurred in (O,rl), and meas(B?,l+l)(r~) = r1. For lXlrl 2 c ~ / L i + ~ , the construction of A n + l could excise a subset from Bn+l(rl), of measure bounded by

( 1 )

( 1 )

( 1 ) 2 -

/ . ( I ) ( 1 )

Denote N1 = N l ( r l ) the least step such that IYlIrf > c/ (LE; ' ) ) , then


2 If we recall that C/L{~+') < IS%lrl 5 c/LE1?',), the result is that

> . meas 2 - CdolJ[l(T-l)/('fl)-l 2(Tp1)/('+1)-1

rl

Thus we ask that 0 < p < 2(7 - 1)/(7 + 1 ) - 1 . In order to make the choice of constants uniform in h, also ask that d ~ l Z l ( ~ - ~ ) / ( ' + ~ ) - ~ < 1 . This is achieved with the present choices of induction parameters do = Lo', IXJ 2 h" as long as r ] + V ( ( T - 1)/(7 + 1 ) - 1) > 0.

involves the excision of 6,+1/Li neighborhoods of the zero sets Zj discussed in Section 4. These intersect the R axis at points we/rn, (4,rn) E B,+l\B,. Using the hypothesis of full nonresonance, if

(2) The construction of

(2) then we retain the point (p,wl) E Nn+l throughout the induction. This requires the choice Z 5 a. We also ask that Z i + 1 2 7, so that the critical excisions of this construction will always fall within ,A'":; thus the parabolic estimates (4.23) will hold in the relevant neighborhood of w1 .

The zero set Zi may intersect the surface closer to w1 than we/m, however these must be at points where W I + lX/2l llp1I2 2 Iwe/rnl - C1 llp112/L?. That is, the points of intersection must satisfy

We ask that C1Li2 < IXl/2. Define the set

(2) (2) gn+l ( r I ) = (0 < r < ri 5 ro; llpll = r , and (p,R(p)) E Nn+I}.

(2) If Y: < (1x1 + l)-' ( c 2 / L z ) ) - 6,+1/L?, then in constructing there is no excision within gnfl. Thus we will select Z so that 6,+1/Li 5 c2/(2Ln+l ). In particular we require that Z 5 a. Let N2 = Nz(r1) be the first induction step at which excisions may occur within (0, r1). We now estimate the measure of the set (0, rl)\ n n z N z 9,+, as in the first case.

(2) (E+2)

(2)

meas ((o,rI)\ nnm2 g 2 1 ~ )

nLN: J ,,Ic2


Using that C/L)Y",++'; S rf 5 C/L::*', the result is that

Thus we also require that

0 < p < 2((a - 2 p - l)/(Z + 2) - 1 ) .

This finishes the proof of the theorem.

Acknowledgments. The authors would like to thank the Dipartement de Physique ThCorique, UniversitC de Genkve, the IHES-Bures sur Yvette, MSRI-Berkeley, and Oxford University for their hospitality, and the National Science Foundation (Grants No. DMS-8920624, No. DMS-9002059, and No. DMS-9203359) for their support of our research. Additionally the first author is supported by a fellowship from the Alfred P. Sloan Foundation. We also thank Percy Deift, Thomas Kriecherbauer, and an anonymous referee for very helpful comments about this paper.

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Received December 199 1 . Revised January 1993.

newton's method and periodic solutions of nonlinear wave equations

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