joceline lega & alain goriely- pulses, fronts and oscillations of an elastic rod

8/3/2019 Joceline Lega & Alain Goriely- Pulses, fronts and oscillations of an elastic rod

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Pulses, fronts and oscillations of an elastic rod

Joceline Lega & Alain Goriely

Department of Mathematics and Program in Applied Mathematics,University of Arizona, Building #89, Tucson, AZ 85721

e-mail: lega,[email protected]

March 9, 1999

Abstract

Two coupled nonlinear Klein-Gordon equations modeling the three-dimensional dynamics of a twisted

elastic rod near its first bifurcation threshold are analyzed. First, it is shown that these equations are

Hamiltonian and that they admit a 2-parameter family of traveling wave solutions. Second, special

solutions corresponding to simple deformations of the elastic rod are considered. The stability of such

configurations is analyzed by means of two coupled Nonlinear Schrodinger equations, which are derived

from the nonlinear Klein-Gordon equations in the limit of small deformations. In particular, it is shown

that periodic solutions are modulationally unstable, which is consistent with the looping process observed

in the writhing instability of elastic filaments. Third, numerical simulations of the nonlinear Klein-Gordon

equations suggesting that traveling pulses are stable, are presented.

KEYWORDS: Elastic Rods, Amplitude equations, Nonlinear Klein-Gordon equations, Nonlinear Schro-

dinger equations.

1 Introduction

The study of elastic deformations in rods has a long tradition in mathematics, physics and engineering, dating

back to Euler. The main challenge is to understand the different possible shapes elastic rods can conforminto under a variety of external stresses. The classical approach to solve such problems is to consider thedifferent static equilibria of rods and study through variational principles the bifurcation between them [1, 2].The dynamical problem, namely how different configurations evolve in time or change to one another is notusually addressed. Another, closely related, classical problem in the theory of elasticity is the propagation ofwaves in elastic media. In thin elastic rods, it is known that two types of waves can exist: they are flexuraland torsional waves and propagate variation of curvature and twist along the rod. Roughly speaking, theformer can be obtained by suddenly moving one end of a long straight rod while the latter can result froma sudden twist of one end of the filament. Both aspects, structural deformations under stress and wavepropagation, can be studied together by considering the dynamics of elastic rods. A convenient setting forsuch a study is provided by the Kirchhoff model for thin elastic filaments.

In this paper we study some aspects of the dynamics of elastic rods. As shown in [3], the Kirchhoff modelcan be used to analyze the stability (with respect to change of tension or twist) of straight twisted rods

under tension. Near the first bifurcation point, that is when straight twisted filaments lose their stability,the rod dynamics may be described by two amplitude equations, which were derived by Goriely and Tabor in[3]. These equations take the form of two nonlinear Klein-Gordon equation describing the coupling of twistdensity to amplitude of deformation. This paper is devoted to an analysis of these amplitude equations,and is organized as follows. In Section 2, we briefly describe the Kirchhoff model. The amplitude equationsfor the near-threshold dynamics are given in Section 3, where we also discuss their scalings, symmetriesand Hamiltonian nature. Section 4 is devoted to the analysis of traveling wave solutions and to simpleconfigurations of the elastic filament. Section 5 discusses the reduction of the coupled Klein-Gordon equationsto two coupled nonlinear Schrodinger equations, which are valid in the limit of small deformations. Solutionsto the latter system are then compared to those of the former, and the stability of some simple traveling

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wave solutions is thereby inferred. Section 6 shows numerical simulations of the amplitude equations, whichsuggest that pulses may propagate in a stable fashion. Conclusions are given in Section 7.

2 The Kirchhoff model

The Kirchhoff model describes the dynamics of a free elastic filament in three dimensions. Assuming that thefilament is thin (its cross section is much smaller than its length) and weakly bent (its curvature radius is, at

all points, much shorter than its length), a one-dimensional theory can be derived where forces and momentsare averaged over cross sections perpendicular to the central axis of the filament. This allows to describe thedynamics in terms of two independent variables, namely time and arc-length (respectively, x and t). Theforces and moments exerted on a cross section of the filament are then written in a local basis, attachedto the central axis XXX= XXX(x, t), which is similar to the Frenet frame. This director basis {d1, d2, d3} isa right-handed orthonormal basis built by taking d3 as the tangent vector to the axis, d3(x, t) = dXXX/dt,and d1 as the vector following the elastic twist in the plane normal to the central axis. Since the curveXXX= XXX(x, t) can be obtained by integrating the tangent vector d3(x, t), i.e. XXX(x, t) =

sd3(s, t)ds, the

space and time evolution of the director basis specifies the kinematics of the filament.As a consequence of the orthonormality of the director basis {d1, d2, d3}, we can write

di = di, di = di i = 1, 2, 3. (1)where ( ) and ( )

stand, respectively, for derivatives with respect to time and arc-length. The vectors and

are called the twist and spin vectors. They can be written as =3

i=1 idi and =3

i=1 idi. Thedynamics of the filament depends on the total force and moment, F = F(s, t) and M = M(s, t), experienced

by the elastic rod. In terms of the director basis, these vectors read F =3

i=1 fidi, M =3

i=1 Midi. TheKirchhoff model combines conservation of linear and angular momentum with the constitutive relationship oflinear elasticity, which relates moments to strains, characterized by the twist vector. For a naturally straightrod (i.e., without intrinsic curvature or twist) with circular cross-section, the scaled Kirchhoff equations read[4, 5]:

F = d3, (2.a)

M + d3 F = d1 d1 + d2 d2, (2.b)M = 1d1 + 2d2 + 3d3, (2.c)

where = 1/(1+ ) ( is the Poisson ratio) measures the ratio between the bending and twisting coefficients

of the rod. In typical elastic material the Poisson ratio varies between 12 and 0, that is, varies between23 and 1. Equations (2), together with (1), can be reduced to a set of 9 second order equations in space andtime, for the 9 unknowns (, , F).

3 Amplitude equations, scalings and Hamiltonian formalism

3.1 Amplitude equations

The stability of a given stationary filament parameterized by ((0), (0), f(0)) can be obtained by expandingall quantities up to first order in a small perturbation parameter and requiring that the basis remainsorthornormal to each order [6]:

di = d(0)i +

d(0)i + O(

2), (3.a)

= (0) +

((1)) + (0) (1)

+ O(2), (3.b)

= (1) + O(2), (3.c)

F =i

f(0)i +

f(1)i + ( f(0))i

d(0)i + O(

2). (3.d)

Inserting this expressions in the Kirchhoff equations and collecting the terms to first order in , a linearequation can be obtained for m = ((1), f(1)):

LE((0), f(0)).m = 0, (4)

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where LE is a linear, second-order differential operator in s and t, whose coefficients depend on s through theunperturbed solution ((0), f(0)). Its explicit form can be found in [7]. For the case of the twisted straightrod under tension, the stationary solution is simply given by

(0) = (0, 0, ) , f(0) =

0, 0, P2

. (5)

Solutions to (4) can be expressed as the sum of fundamental modes of the form:

m = et

Aneinx + A

neinx

, (6)

where C6 and the growth rate is related to the mode n by the dispersion relation, (, n) = 0,obtained by substituting (6) into (4). The neutral curves are the curves in parameter space for whichstationary solutions exist. They are obtained by solving (0, n) = 0, which reads:

2 n2 2( 1) P2 n22 2( 2)2n2 = 0. (7)

The analysis of this dispersion relation reveals that the straight infinite twisted pulled rod becomes unstablefor c = 2P/. This relation is known in elasticity theory as Loves criterion. For this value, a stationarysolution can be written as

m = B0 + Anceincx + Ance

incx + . . . , (8)

where the dots stand for higher order corrections and nc = P(2 )/,

0 = (0, 0, 1, 0, 0, 1) , nc =

1, i, 0,iP2, P2, 0 . (9)For fixed P, the straight rod becomes unstable at the critical twist c and evolves towards a configuationwhich, to first order, is a helical filament and reads

XXX=

x,2A

Pcos xP,

2A

Psin xP

+ . . . . (10)

Near threshold, the nonlinear dynamics of the filament is described by the following amplitude equations[7]:

P2

+ 1P2

2

At2

2

Ax2

= PA

E 2P|A|2 + Bx

, (11.a)

2

2B

t2

2B

x2= 2P|A|

2

x. (11.b)

In this nonlinear analysis, B = B(x, t) and A = A(x, t) represent, respectively, the slowly-varying amplitudesof the axial twist and the unstable helical mode. For given solutions A and B of (11), the shape and motionof the filament in space is obtained by integrating the tangent vector. To second order, the filament shapeis given by:

XXX(x, t) =

(1 2 |A|2)dx

[ cos(P x) (2 Re(A) + B Im(A)) sin(P x) (2 Im(A)B Re(A))] dx[ sin(P x) (2 Re(A) + B Im(A)) + cos(P x) (2 Im(A)B Re(A))] dx

. (12)

The rest of this paper is devoted to the analysis of these solutions and its consequences for the dynamics ofrods.

3.2 Scalings

To keep the discussion as general as possible and to reduce the number of free parameters, it is convenientto re-write the amplitude equations in terms of scaled quantities. If we make the changes of variables

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t =

2

L t, x = Lx, A =

P2 + 1

(2LP2)A, B =

(P2 + 1)

(2P3L)B, (13)

Eqns. (11) become, after dropping the primes,

2A

t2

c20

2A

x2

= A

|A

|2A +

B

x

A, (14.a)

2B

t2

2B

x2= |A|

2

x, (14.b)

where

c20 =2P2

(P2 + 1)and =

2EP3L2

P2 + 1. (15)

Only two parameters are left. The first one, , measures the linear growth rate of perturbations to thesolution A=0 in the absence of B. The second one, c0, is the ratio vA/vB of the group velocities vA andvB describing the speed at which perturbations of A and B propagate. These scaled equations provide aconvenient way of analyzing the dynamics and the solutions of Eqns. (11), and most of the discussion belowwill refer to them. The new space scale depends on L, a characteristic length scale of the problem. For a

finite rod, we take L as the total length of the elastic filament. In what follows, we refer to these equations asthe coupled nonlinear Klein-Gordon equations (CNLKG). Similar coupled nonlinear Klein-Gordon equationsappear in a variety of fields such as plasma physics (See [8]; p.318), partial difference equations [9], baroclinicinstabilities [10], or more generally as amplitude equations for dispersive systems [11].

3.3 Hamiltonian form

From the scaled form (14), it is straightforward to see that the coupled Klein-Gordon equations can bewritten in Hamiltonian form

A

t=

H

u,

u

t= H

A, (16.a)

B

t=

H

v,

v

t= H

B, (16.b)

where refers to the Frechet derivative,

u =A

t, v =

B

t, (17)

and H =

h(x, t) dx with

h(x, t) = c20

Ax2

|A|2 + 12|A|4 |A|2 B

x+

At2

+1

2

B

x

2+

1

2

B

t

2. (18)

The integral is taken over the real line if h(x, t) is L1[(, +)]. If A does not vanish at infinity, theintegration is taken on a suitable box (for instance, one period of the solution for periodic solutions).

3.4 Symmetries

Equations (14) have a few symmetries that can be used to simplify our analysis of special solutions. Below,we assume that a0(s, ) and b0(s, ) are solutions to (14). The solutions A(x, t) and B(x, t) are obtained byapplying the indicated symmetries to a0 and b0.

1. Space translation invariance:

A(x, t) = a0(x + x0, t), B(x, t) = b0(x + x0, t), (19)

where x0 is a real arbitrary constant.

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2. Time translation invariance:

A(x, t) = a0(x, t + t0), B(x, t) = b0(x, t + t0), (20)

where t0 is a real arbitrary constant.

3. Uniform shift on B

A(x, t) = a0(x, t), B(x, t) = b0(x, t) + , (21)where is a real arbitrary constant. This transformation corresponds to a uniform rotation of the localframe of reference (d1, d2, d3) about the tangent vector to the elastic rod d3, and is not physicallyrelevant.

4. Gauge invariance:

A(x, t) = a0(x, t) exp(i), B(x, t) = b0(x, t), (22)

where is a real arbitrary constant. This transformation corresponds to a uniform translation of theperiodic solution of envelope A and is due to the invariance of the Kirchhoff equations with respect tospace-time translations before bifurcation.

5. Space reflection symmetry

A(x, t) = a0(x, t), B(x, t) = b0(x, t). (23)

6. Time reflection symmetry

A(x, t) = a0(x,t), B(x, t) = b0(x,t), (24)which is due to the fact that the original system is time reversible.

7. Dilation invariance:

A(x, t) = a0 (x, t) , B(x, t) = b0 (x, t) +

2 1 x, (25)

where is a real arbitrary constant. This transformation is due to the fact that adding a linear termin x to B renormalizes the growth rate , which in turn can be changed back to its initial value byscaling space, time, and the envelopes A and B.

Whereas the Hamiltonian is preserved by transformations 1 to 6, the dilation 7 does not conserve H.Since the amplitude equations (11) have been derived by assuming that the solution m defined in Section 2remains bounded at infinity, we will set so that B tends to a constant at infinity. This condition turns outto be necessary for H to remain bounded.

4 Traveling waves solutions of CNLKG

We now look for a 2-parameter family of solutions of the form:

A = a()eit, B = b(), (26)

where = x ct. This family includes periodic stationary solutions (c = = 0), oscillatory periodicsolutions (c = 0), homogeneous solutions (a, b constant, c = = 0), plane waves (b, and |A| constant),non-oscillating traveling waves solutions and pulses ( = 0). The general case where both c and do notvanish correspond to traveling waves. These solutions periodically oscillate in time as they travel at constantspeed c. Together with (26), Eqns. (14) read (where primes denotes -derivatives)

(c2 c20)a + 2ica = a(2 + |a|2 + b), (27.a)(c2 1)b = (|a|2). (27.b)

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The last equations can be integrated once to yield (c2 1)(bK1) = |a|2. We then use the expression forb to simplify (27.a)

(c2 c20)a + 2ica = a

2 + + K1 c2

(c2 1) |a|2

(28)

In order to obtain a set of real differential equations, we set a = R()ei(), which gives

(c2 c20)R = R

2 + + K1 2c + (c2 c0)22 c2(c2 1) R

2

, (29.a)

2

c

(c2 c20)

R + R = 0, (29.b)

where = . The system for (R, ) has two first integrals:

H =1

2(R)2 +

1

2

2 ( + K1 +

2)

c

R2 +

c2R4

4(c2 1)(c2 c20), (30.a)

G =

(c2 c20) c

R2. (30.b)

The first integral G is reminiscent of Keplers second law for the motion of a particle in a central force field.

It can be used to simplify H as follows:

H1 = H+ G2(c2 + c20) ( + K1)(c2 c20)

2c(c2 c20)2

=1

2(R)2 +

G2

2(c2 c20)2R2 +

2c20 ( + K1)(c2 c20)2(c2 c20)2

R2 +c2

4(c2 1)(c2 c20)R4.

(31)

Thus, H1 is an Hamiltonian describing the motion of a particle in a one-dimensional effective potentialVeff = H1 (R)2/2 = v2R2 + v2R2 + v4R4. This reduced Hamiltonian system can be solved through thechange of variables

= , z = v4R2, (32)

which transforms the Hamiltonian H1 into the standard form:dz

dt

2=

4

z3 z1 (z z1)(z2 z)(z3 z), (33)

where 1 =

2(z3 z1) andz1 + z2 + z3 = v2, z1z2 + z2z3 + z3z1 = v4H1, z1z2z3 = v24v22. (34)

The general solution of (33) is z = z1 + (z2 z1)sn2(|k) where k2 = (z2 z1)/(z3 z1). That is,

R =z1 + (z2 z1)sn2( |k)

v4

. (35)

The conditions for the roots z1, z2, z3 to be real is:

27v42v24 + 2H1(9v2v

22 2H21 )v4 v22H21 + 4v32v22 < 0. (36)

The first integral G can then be used to give an explicit form of the angular coordinate in terms ofincomplete elliptic integrals of the third kind. Explicit forms for the periods of given periodic orbits can beobtained in a similar fashion.

To understand the different types of orbits that the system H1 may exhibit it is easier to study thepotential Veff directly. There are two main cases that can be distinguished depending on whether G vanishes

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v2>0v4>0

v20

v2>0v40v4>0

a. b.

c. d.

v20

v2>0v4


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0.7

0.8

0.9

1

1.1

1.2

1.3

-3 -2 -1 0 1 2 3

R

Figure 3: A traveling hole solution (v2 = 4, v4 = 1, v2 = 5, c0 = 3, = 0)

or not. In each cases, four different potential shapes can be obtained (depending on the sign of the coefficients

v2, v4). These potentials are shown, respectively, on Fig. 1 and Fig. 2.We now focus on localized solutions, that is solutions for which the amplitudes A and B tend asymp-totically to a constant and vary rapidly in a finite region. At the level of the Hamiltonian system withHamiltonian H1, we look for homoclinic and heteroclinic orbits connecting the fixed point R0 to himself(homoclinic case) or to R0 (heteroclinic case). In both cases, we choose the constant K1 to ensure thatlim b = 0, that is K1 = R

20/(c

2 1). Analytic expressions will be given with a minimum number offree parameters. More general solutions can be obtained by using the symmetries discussed in Section 3.4.

4.1 Traveling holes

If G = 0, the only possible localized solutions occur for v2 > 0 and v4 < 0. These orbits are, for theHamiltonian H1, homoclinic to the fixed point R0 (See Fig. 1c). In this case, one sets 0 < z1 < z2 = z3 =v4R20 in (35) to obtain:

R2 = R20 2 sech2(), (37)

where

2 = 6v4R20 2v2, and 2 =(3v4R

20 + v2)

v4. (38)

The fixed point R0 is the largest root of 2z6 + v2z

4 v22 = 0. The homoclinic orbit exists when H1 =R20(2v2 + 3v4R

20). Explicit forms of = () and b = b() can be obtained in terms of elementary functions.

A typical traveling hole amplitude profile is shown in Fig. 3.

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Re(A)

B

| A|

x

Figure 4: A traveling pulse ( = 1/4, c = 2, c0 =

2, = 1/4, = 3/4)

c

Figure 5: Filament corresponding to the homoclinic solution of Figure 3.

4.2 Traveling pulses

Traveling pulses exist when G = 0, v2 < 0 and v4 > 0 (Fig. 2b). They correspond to orbits of the reduced

Hamiltonian system which are homoclinic to the fixed point R0 = 0, at which H1 = 0. They read

a = sech()exp

i

c

c2 c20

. (39.a)

b =2

(1 c2) tanh(), (39.b)

where

2 =2(c2 1)

c2(c2 c20)

(c2 c20) 2c20

and 2 =(c2 c20) 2c20

(c2 c20)2. (40)

This solution exists whenever 2 > 0 and 2 > 0. Typical profiles of A and B are shown in Fig. 4, and thecorresponding filament is depicted in Fig. 5.

4.3 Fronts

There exists yet another localized solution when G = 0. If v2 > 0 and v4 < 0 the Hamiltonian H1accommodates heteroclinic orbits (See Fig. 2d), given by

a = tanh()exp

i

c

c2 c20

, (41.a)

b =2

(c2 1) tanh(), (41.b)

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where T = t, X2 = 2x, and is a small parameter. As shown in Appendix A, the amplitudes A and B can

be written, to order 3, as

A = F1(x, t)e

i

c0x

+ F2(x, t)e

i

c0x

, (49.a)

B = +ic0

2

F1(x, t)F

2 (x, t)e

2i

c0x F2(x, t)F1 (x, t)e2i

c0x

. (49.b)

The envelopes F1 and F2 satisfy two solvability conditions obtained at order 3

(see Appendix A). They takethe form of two coupled nonlinear Schrodinger equations for F1 and F2, which read

2F1t2

+ 2ic0

F1x

+ |F1|2F1 + |F2|2F1 = 0, (50.a)2F2t2

2ic0 F2x

+ |F2|2F2 + |F1|2F2 = 0. (50.b)

These equations, known as the Manakov equations [13, 14], have the symmetry F1 F2 and F2 F1 .As shown in [14], they are completely integrable and admit a Lax Pair. Their solitary wave solutions areclassified in [15] and various perturbations of these equations have been considered for the description oflight propagation in birefrengent fibers [16] or counterpropagation in Kerr media [17, 18] (see also [19] and[20] as well as references therein).

5.2 Special solutions of the reduced equations

We now consider particular solutions to the two coupled nonlinear Schrodinger equations (50) and comparethem to similar solutions of the coupled Klein-Gordon equations.

5.2.1 Plane waves

Plane wave solutions to (50) of the form

F1 =

2 + 2c0

p exp[i(px + t)], F2 = 0, (51)

correspond to plane wave solutions of CNLKG

A = R exp

i

px + t x

c0

, B = 0. (52)

Indeed, R2 in (52) is given by R2 = 2 + 2c0p

c20p2 = 2 + 2c0p

+ O(p2), which is consistent withthe scaling p = O(2) and the fact that expressions (48) are valid at order 3.

Since plane wave solutions of (50) are modulationally unstable, plane wave solutions of CNLKG are alsounstable. As a consequence, traveling holes and fronts of CNLKG are unstable since their asymptotic states,which are plane wave solutions, are unstable.

5.2.2 Stationary solutions

Stationary solutions to (50) with F1 = R exp(iqx) and F2 = F1 are such that R

2 = c0q

. In terms ofA and B, and assuming = 0 (which can always be done by virtue of Symmetry 3 of Section 3.4), these

solutions read

A = cos

c0 q

x

+ O(4), (53.a)

B = sin

2

c0 q

x

+ O(4), (53.b)

where

2 = 4 c0 q

, = c20 q. (54)

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If we translate this solution by a quarter of its period (which again is allowed by Symmetry 1 of Section3.4), we obtain

A = cos

c0 q

x +

2

+ O(4) = sin

c0 q

x

+ O(4), (55.a)

B = sin

2

c0 q

x +

+ O(4) = sin

2

c0 q

x

+ O(4). (55.b)

This is exactly the near-threshold expansion (46, 47) of the periodic solutions (45). Therefore, the lattercorrespond to solutions of constant amplitude of Eqns. (50), with F2 = F

1 . Since, with F2 = F

1 , the two

coupled equations (50) reduce to the self-focusing nonlinear Schrodinger equation

2F1t2

+ 2ic0

F1x

+ 2|F1|2F1 = 0, (56)

the periodic solutions (45) are modulationally unstable near threshold. This suggests that amplitude defor-mations will focalize in finite time, and is consistent with the results obtained in [21] for the linear stabilityanalysis of the infinite helix which appears above the writhing bifurcation of a twisted elastic rod.

5.2.3 Traveling pulses

Soliton solutions of (50) of the form

F1 =

2c0c

sech

c0

c2(x ct)

exp

i

c0

ct

, F2 = 0, (57)

correspond to solutions of (48) with

A = sech

(x ct)

exp

i

k(x ct) + t

, B = 0, (58)

and

=c0

2

c, =

c0c2

, k =

c0, =

c20 c2c0c

. (59)

Since |A|

2 c0

cmust be small for the coupled nonlinear Schrodinger equations to be valid, solution (57)

are valid in the limit of large c. More precisely, since c0 and

are finite, c should be such that c1 = O().We now compare this solution to traveling pulse solutions of CNLKG. The latter are of the form:

A = sech((x ct))exp[i (k(x ct) + t)] , (60.a)B =

2

(c2 1) tanh[(x ct)] , (60.b)

where

2 = 2(c2 1)

c2 2

2(c2 1)c20c2(c2 c20)

, 2 =

c2 c20 c

20

2

(c2 c20)2(61.a)

k = cc2 c20

. (61.b)

With = =

c20 c2

c0c, the last equation becomes k =

c0= k, and 2 and 2 read

2 = 2c20c2

1 1

c2

= 2 + O(

1

c4) = 2 + O(4), (62.a)

2 =c20

c4+ O(

1

c6) = 2 + O(6). (62.b)

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Figure 10: Shape of the filament corresponding to a double soliton solution of the coupled nonlinearSchrodinger equations. The parameters used to plot this figure, given in terms of the unscaled equations,are = 0.75, P = 3, E= 0.1, L = 40, = 0, and c = 2.

In other words, the expressions of 2, 2 and 2 and 2 are consistent to lowest order. The fact that the

nonlinear Schrodinger equations give B = 0 instead of B =2

(c2 1) tanh[(x ct)] can be understood asfollows. Traveling pulse solutions of CNLKG are found by solving a differential equation for A and then by

solving Eqn. (27.b) for b. This amounts to solving

B =dB

d= |A|

2

c2 1 = 2

c2 11

cosh2(), (63)

where = x ct. But with the expression of 2, we have2

c2 1 =2c20

c4= O(4), (64)

so that B = 0 at order 3, i.e. B = = constant at order 3. Therefore, we conclude that soliton solutions(57) of the coupled nonlinear Schrodinger equations correspond to traveling pulses (39) of CNLKG.

5.2.4 Double solitons

The coupled nonlinear Schrodinger equations (50) also admit solutions of the form

F1 =

c0c

sech

c0

c2(x ct)

exp

i

c0

ct

, F2 = F1, (65)

which correspond, in terms of A and B, to

A = 2

c0c

sech

c0

c2(x ct)

cos

c0

x c

20

ct

, (66.a)

B =

c30c2

sech

c0

c2(x ct)

sin

2

c0

x c

20

ct

. (66.b)

Figure 10 shows a reconstructed filament with A and B given above.

6 Numerical results

Following the stability analysis presented in this paper, we conclude that traveling holes, fronts and periodicsolutions of the coupled Klein-Gordon equations are unstable, at least in systems which are large enoughfor the modulational instability to develop. On the other hand, traveling pulses are unstable if is positivesince their asymptotic states (A = 0, B = ) experience exponential growth, but may be stable for negativevalues of. In this section, we show results of numerical simulations of the original (unscaled) coupled Klein-Gordon equations (11), which suggest that traveling pulses are stable in some parameter regimes. In order to

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Figure 11: Numerical simulation of the original (unscaled) nonlinear Klein-Gordon equations, showing thestable propagation of a traveling pulse. Parameters used in the simulation are = 0.75, P = 3, E = 0.1,

L = 40, = 0, and c = 0.6124.

study traveling waves, we designed a code with non-reflecting [22] boundary conditions, which minimize theamount of reflection at the boundaries. The code is second order in time (a typical time-step is dt = 0.01)and uses spectral-like finite differences [23] (a typical box length value is L = 40, which corresponds to amesh size dx = 0.1). Although no symplectic structure is implemented in the numerics, we checked that theHamiltonian

H =

dx

P2

P2 + 1

Ax2

+P4

P2 + 1|A|4 P

3

P2 + 1|A|2

B

x+ E

+

At2

+P2

4(P2

+ 1)

B

x

+ E2

+P2

2(P2

+ 1)

B

t2

, (67)is conserved to within 0.001% of its value. The simulation has a graphical interface, built with the AdvancedVisual Software AVS 5, which allows for interactive change of the parameters.

Figure 11 illustrates the stable propagation of a traveling pulse (see (39)), which, in terms of unscaled

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variables, reads

A =

P2 + 1

2LP2sech

L

x c

2t

exp

i

k

L

x c

2t

+

L

2t

, (68.a)

B =2

(c2 1)P2 + 1

2LP3tanh

L

x c

2t

, (68.b)

where

2 =2(c2 1)

c2(c2 c20)

(c2 c20) 2c20

, 2 =(c2 c20) 2c20

(c2 c20)2, (69.a)

c20 =2

P2

P2 + 1, =

2EP3L2

P2 + 1, k =

c

c2 c20. (69.b)

The parameter values are given in the caption of Fig. 11. We only show the reconstructed filament, whoseshape is computed from the numerical values of A and B according to formula (12) which is accurate tosecond order in the perturbation parameter.

Our numerical results therefore suggest that traveling pulses are likely to propagate in a stable fashion, atleast in some parameter regime. A complete answer to this question would require an analysis of the space-and time-dependent Klein-Gordon equations linearized about a traveling pulse, and is beyond the scope of

this paper.

7 Conclusions

This paper gives a detailed description of the dynamics of the coupled nonlinear Klein-Gordon equations,which model an elastic filament subjected to external twist in the vicinity of the first bifurcation threshold.It is shown first that these equations have only two relevant parameters, namely the linear growth rate ofamplitude deformations and the ratio of the group velocities of amplitude and twist modulations, and second,that they are Hamiltonian. A complete description of traveling wave solutions to these equations is presented,and localized solutions are identified. Because such solutions are of physical interest, the question of theirstability must be addressed. This is accomplished by means of two coupled nonlinear Schrodinger equations,valid in the limit of small amplitude deformations. It turns out that these equations, known as Manakovs

equations, are integrable and arise as a limiting case of model equations describing propagation of light inbirefringent fibers or counterpropagating beams in Kerr media. Moreover, because Manakovs equation aredefocusing, their plane wave solutions are modulationally unstable and, as a consequence, traveling holesand fronts of the coupled Klein-Gordon equations have unstable asymptotic states. Interestingly, periodicsolutions, which correspond to modulated helical filaments, are also proven to be unstable, and this self-focusing instability is consistent with the looping process observed in elastic rods subject to excessive twist.It was shown in [24] that, when made unstable, infinite twisted rods tend to first localize deformations andthen form twisted loops (eventually giving rise to a braided structures). The analysis of the periodic solutionsof CNLKG therefore suggests that the formation of loops in finite rods results from a self-focusing instability.Finally, our numerical simulations show that traveling pulses are stable in some parameter regime, but onlyif the system is kept below threshold.

Acknowledgments

The authors would like to thank N. Ercolani and M. Tabor for many interesting discussions. This work ispartly supported by Nato Collaborative Research Grant NATO-CRG 97/037.

A Derivation of the nonlinear Schrodinger equations

This section is devoted to a multiple scales analysis of time-independent periodic solutions to the scaledcoupled Klein-Gordon equations (14). As explained in Section 5, we set

A(x, t) = A1(x, T, X2) + 2A2(x, T, X2) +

3A3(x, T, X2) + . . . (70.a)

B(x, t) = B1(x , T , X 2) + 2B2(x, T, X2) +

3B3(x, T, X2) + . . . (70.b)

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where T = t, X2 = 2x, is a small parameter, and the dots stand for higher order corrections. After

substituting in (14) and equating like powers in , we get a hierarchy of partial differential equations whichcan be solved at each order. At order , we obtain

c02

2

x2A1(x, T, X2) + A1(x, T, X2) = 0, (71.a)

2

x2B1(x, T, X2) = 0, (71.b)

which gives

A1(x, T, X2) = F1(T, X2)e

ix

c0 + F2(T, X2)eix

c0 , (72.a)

B1(x, T, X2) = , (72.b)

where F1 and F2 are two complex envelopes and is an arbitrary constant. Only the terms which do notlead to divergence at infinity have been kept in B1.

At order 2, we get

A2(x, T, X2) + c02

2

x2A2(x, T, X2) =

xB1(x, T, X2)

A1(x, T, X2) = 0, (73.a)

2

x2B2(x, T, X2) =

xA1(x , T , X 2)A

1(x , T , X 2) + A1(x , T , X 2)

xA1(x, T, X2). (73.b)

After substituting in the expressions for A1 and its complex conjugate, we can solve for A2 and B2 andobtain

A2(x, T, X2) = 0, (74.a)

B2(x, T, X2) =ic0

2

e2

ix

c0 F2 (T, X2)F1(T, X2) e2ix

c0 F2(T, X2)F1 (T, X2)

. (74.b)

We do not include in A2 and B2 terms which are in the kernel of the linear operator obtained at order .These terms have already been taken into account in the expressions of A1 and B1.

At order 3, the Klein-Gordon equations read

c02

2

x2A3(x, T, X2) + A3(x, T, X2) (75.a)

=

2

T2 F1(T, X2) + |F1(T, X2)|2 F1(T, X2) + 2ic0 X2 F1(T, X2) + |F2(T, X2)|2 F1(T, X2)

ei

x

c0

+

2

T2F2(T, X2) + |F2(T, X2)|2 F2(T, X2) 2ic0

X2F2(T, X2) + |F1(T, X2)|2 F2(T, X2)

ei

x

c0 ,

2

x2B3(x, T, X2) = 0. (75.b)

In order to avoid secular terms in A3, we need to impose the two solvability conditions

2

T2F1(T, X2) + |F2(T, X2)|2 F1(T, X2) + 2 ic0

X2F1(T, X2)

+ |F1(T, X2)|2 F1(T, X2) = 0 (76.a)2

T2F2(T, X2) +

|F1(T, X2)

|2 F2(T, X2)

2 ic0

X2F2(T, X2)

+ |F2(T, X2)|2 F2(T, X2) = 0, (76.b)which are two coupled nonlinear Schrodinger equations for the envelopes F1 and F2. If we re-write these equa-

tions in terms of the original variables x and t and define F1(x, t) = F1(t,2x) and F2(x, t) = F2(t,

2x),we get, after dropping the primes,

2F1t2

+ 2ic0

F1x

+ |F1|2F1 + |F2|2F1 = 0 (77.a)2F2t2

2ic0 F2x

+ |F2|2F2 + |F1|2F2 = 0. (77.b)

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If these equations are satisfied, we can then choose A3(x, T, X2) = 0 and B3(x, T, X2) = 0. To order 3, the

solutions A and B are then given by

A(x, t) = F1(t, x)e

ix

c0 + F2(t, x)eix

c0 + O(4) (78.a)

B(x, t) = +ic0

2

e2

ix

c0 F2 (t, x)F1(t, x) e2ix

c0 F2(t, x)F

1 (t, x)

+ O(4), (78.b)

where we have set

= and then dropped the prime.

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[18] M. Haelterman, A.P. Sheppard, and A.W. Snyder, Bimodal counterpropagating spatial solitary-waves,Optics Communications 103, 145-152 (1993).

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joceline lega & alain goriely- pulses, fronts and oscillations of an elastic rod

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