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Symmetry reductions, exact solutions, and conservation laws of thegeneralized Zakharov equationsEerdun Buhe and George W. Bluman Citation: Journal of Mathematical Physics 56, 101501 (2015); doi: 10.1063/1.4931962 View online: http://dx.doi.org/10.1063/1.4931962 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/56/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Conservation Laws of the Self‐adjoint K(m,n) Equation with Generalized Evolution Term AIP Conf. Proc. 1389, 1372 (2011); 10.1063/1.3637876 Multidimensional partial differential equation systems: Nonlocal symmetries, nonlocal conservationlaws, exact solutions J. Math. Phys. 51, 103522 (2010); 10.1063/1.3496383 On the algebraic aspect of singular solutions to conservation laws systems AIP Conf. Proc. 834, 206 (2006); 10.1063/1.2205804 Conservation Laws, Symmetries, and Elementary Particles Phys. Teach. 43, 266 (2005); 10.1119/1.1903808 Lie groups and differential equations: symmetries, conservation laws, and exact solutions ofmathematical models in physics Phys. Part. Nucl. 28, 241 (1997); 10.1134/1.953039
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JOURNAL OF MATHEMATICAL PHYSICS 56, 101501 (2015)
Symmetry reductions, exact solutions, and conservationlaws of the generalized Zakharov equations
Eerdun Buhe1,2,3,a) and George W. Bluman2,b)1Department of Mathematics, Hohhot University for Nationalities, Hohhot 010051, China2Department of Mathematics, University of British Columbia, Vancouver,British Columbia V6T 1Z2, Canada3School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China
(Received 24 October 2014; accepted 16 September 2015; published online 5 October 2015)
In this paper, the generalized Zakharov equations, which describe interactions be-tween high- and low-frequency waves in plasma physics are studied from the perspec-tive of Lie symmetry analysis and conservation laws. Based on some subalgebrasof symmetries, several reductions and numerous new exact solutions are obtained.All of these solutions represent modified traveling waves. The obtained solutionsinclude expressions involving Airy functions, Bessel functions, Whittaker functions,and generalized hypergeometric functions. Previously unknown conservation lawsare constructed for the generalized Zakharov equations using the direct method.Profiles are presented for some of these new solutions. C 2015 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4931962]
I. INTRODUCTION
It is well known that nonlinear evolution equations (NLEEs) are related to nonlinear phenom-ena in various fields. In understanding these phenomena, it is of paramount importance to seek exactsolutions. One is able to find explicit solutions only for special cases. Nonetheless, in recent yearsvarious methods of solving these types of NLEEs have been proposed, including Hirota’s bilinearmethod,1 the inverse scattering method,2 use of Bäcklund transformation3 or Darboux transforma-tion,4 truncated Painlevé expansion,5 Lie symmetry method,6–9 the homogenous balance method,10
the variational iteration method,11 and the semi-inverse method.12
The Lie symmetry method plays a significant role in seeking solutions for nonlinear differentialequations. It is based upon the study of the invariance of differential equations under one parameterLie groups of point transformations,6–9 and it systematically unifies and extends well known ad hoctechniques to construct explicit solutions for differential equations, especially for nonlinear partialdifferential equations (PDEs).
Conservation laws play an essential role in the study and solution of boundary value prob-lems for DEs. In the last few decades, active research efforts have been made on the derivationof conservation laws for DEs. Significant methods have been developed for the construction ofconservation laws, such as Noether’s theorem13 for variational problems, direct method (multi-plier approach),14,15 symmetry action on a known conservation law,16 and linearizing operatorapproach,17,14 etc.
In this work, we consider the generalized Zakharov equations (GZEs)18 in the form
iEt + Exx − 2β |E |2E + 2EF = 0,Ft t − Fxx + (|E |2)xx = 0,
(1)
where E is the complex envelope of the high-frequency electric field, the low-frequency field F isthe plasma density measured from its equilibrium value, and the cubic term in the first equation
a)Electronic mail: [email protected])Electronic mail: [email protected]
0022-2488/2015/56(10)/101501/14/$30.00 56, 101501-1 © 2015 AIP Publishing LLC
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101501-2 E. Buhe and G. W. Bluman J. Math. Phys. 56, 101501 (2015)
of Eqs. (1) describes nonlinear self-interaction in the high-frequency subsystem which correspondsto a self-focusing effect in plasma physics. The GZEs are a universal model of interaction be-tween high- and low-frequency waves in one dimension. When β = 0, (1) reduces to the classicalZakharov equation of plasma physics19 and has an important particular solution — the Langmuirsoliton. Obviously, the GZEs is a strongly nonlinear system and it is quite difficult to obtain itssolitary wave solutions.
Recently, extensive studies have been carried out by many authors on the GZEs. In Refs. 20and 21, the solitary wave solutions of the GZEs have been obtained by the well-known He’s vari-ational approach. Wang and Li22 introduced periodic wave solutions of GZEs using the extendedF-expansion method. In Ref. 23, the Painlevé property and some exact solutions of the GZEs havebeen obtained using the truncated Painlevé expansion. Dai and Xu24 studied the dynamic behaviorsof some exact traveling wave solutions to the GZEs.
In order to study Lie symmetry reductions, exact solutions, and conservation laws of GZEs(1), we express the complex envelope as E(x, t) = u(x, t) + iv(x, t), with real high-frequency wavesu(x, t) and v(x, t). After substituting the complex envelope expression into (1) and separating its realand imaginary parts, we obtain
2uF − 2βu(u2 + v2) − vt + uxx = 0,2vF − 2βv(u2 + v2) + ut + vxx = 0,Ft t − Fxx + (u2 + v2)xx = 0.
(2)
The layout of the rest of this paper is as follows: In Section II, some exact solutions of (2)are obtained by using Lie symmetry reductions along with some algebraic analysis under some Lieoptimal subalgebras. In Section III, conservation laws of (2) are constructed through use of thedirect (multiplier) method. Finally, conclusions are summarized in Section IV.
II. LIE SYMMETRY ANALYSIS OF GZES
The point symmetry group of generalized Zakharov equations (2) is generated by a vector fieldof the form
Γ =
2i=1
ξi∂
∂xα+
3j=1
η j∂
∂uβ, (3)
where the infinitesimals ξi = ξi(xα,uβ), η j = η j(xα,uβ), and (x1, x2) = (x, t), (u1,u2,u3) = (u, v,F).The application of the second extension Γ(2) 6–8 to (2) results in an overdetermined system of linearpartial differential equations for the infinitesimals.
Wu’s method (also called the characteristic-set algorithm)25–28 was established by the Chinesemathematician Wu Wen Tsun in the 1970s, based on Ritt’s theory. The differential analogue of Wu’smethod was proposed in the 1980s.26 The method is especially on target to deal with the zero setof a differential polynomial system and efficient differential elimination without directly involvingthe concept of an algebra ideal. Here, the general solutions of the overdetermined system of linearpartial differential equations with the aid of Wu’s method is given by
ξ1(x, t,u, v,F) = C1,
ξ2(x, t,u, v,F) = C2,
η1(x, t,u, v,F) = −(C3t2 + 2C4t − C5)v,η2(x, t,u, v,F) = (C3t2 + 2C4t − C5)u,η3(x, t,u, v,F) = C3t + C4,
(4)
where Ci, i = 1, . . . ,5 are five arbitrary constants. Hence, the point symmetries of (2) form a fivedimensional Lie algebraG spanned by the following linearly independent operators:
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101501-3 E. Buhe and G. W. Bluman J. Math. Phys. 56, 101501 (2015)
Γ1 =∂
∂t,
Γ2 =∂
∂x,
Γ3 = −t2v∂
∂u+ t2u
∂
∂v+ t
∂
∂F,
Γ4 = −2tv∂
∂u+ 2tu
∂
∂v+
∂
∂F,
Γ5 = v∂
∂u− u
∂
∂v.
(5)
The operators Γ1 and Γ2 are related to the obvious invariance of Eqs. (2) under space and timetranslations, respectively. The point symmetry vector field Γ5 is related to invariance of Eqs. (2)under rotations in u − v space.
A. One-dimensional optimal system of subalgebras
One does not need to use all the one-dimensional subalgebras of Lie algebraG (5) to constructinvariant solutions. However, a well-known standard procedure8 allows one to classify all one-dimensional subalgebras into subsets of conjugate subalgebras. This involves constructing theadjoint representation group, which introduces a conjugate relation in the set of all one-dimensionalsubalgebras. If one uses only one representative from each family of equivalent subalgebras, anoptimal set of subalgebras is created. The corresponding set of invariant solutions is then the min-imal list from which one can obtain all other invariant solutions of one-dimensional subalgebrassimply via transformations. In this subsection, we present the optimal system of one-dimensionalsubalgebras for the Lie algebra for point symmetries (5) of system (2) to obtain the correspondingoptimal set of group-invariant solutions.
Each Γi of basis symmetries (5) generates an adjoint representation (or interior automorphism)Ad(exp(εΓi))Γj defined by Ref. 8,
Ad(exp(εΓi))Γj = Γj − ε[Γi,Γj] + 12ε2[Γi, [Γi,Γj]] − · · ·.
Here, [Γi,Γj] is the commutator given by
[Γi,Γj] = ΓiΓj − ΓjΓi.The commutator table of the Lie point symmetries of system (2) and the adjoint representations
of the symmetry group of (2) on its Lie algebra are given in Tables I and II, respectively. Essen-tially these adjoint representations simply permute amongst “similar” one-dimensional subalgebras.Hence, they are used to identify similar one-dimensional subalgebras. However, before proceedingwith the classification scheme one needs to identify invariants of the full adjoint action as theseplace restrictions on how far one can expect to “simplify” a given arbitrary element,
Γ =
5i=1
CiΓi, (6)
TABLE I. Commutator table of the Lie algebra of system (2).
[Γi,Γj] Γ1 Γ2 Γ3 Γ4 Γ5
Γ1 0 0 Γ4 −2Γ5 0Γ2 0 0 0 0 0Γ3 −Γ4 0 0 0 0Γ4 2Γ5 0 0 0 0Γ5 0 0 0 0 0
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101501-4 E. Buhe and G. W. Bluman J. Math. Phys. 56, 101501 (2015)
TABLE II. Adjoint table of the Lie algebra of system (2).
Ad Γ1 Γ2 Γ3 Γ4 Γ5
Γ1 Γ1 Γ2 Γ3 Γ4 Γ5
Γ2 Γ1 Γ2 Γ3−εΓ4−ε2Γ5 Γ4+2εΓ5 Γ5
Γ3 Γ1 Γ2+εΓ4 Γ3 Γ4 Γ5
Γ4 Γ1 Γ2−2εΓ5 Γ3 Γ4 Γ5
Γ5 Γ1 Γ2 Γ3 Γ4 Γ5
of the GZEs Lie algebra G. A real function η :G → R defined by η(Γ) = ψ(C1,C2,C3,C4,C5) forsome other function ψ is an invariant of the full adjoint action in Table II if for each nonzero Γ ∈ G,
η(Ad(exp(εΓi))Γ) = η(Γ), i = 1, . . . ,5.
Note that elements of G can be represented by vectors A = (C1, . . . ,C5) ∈ R5 since each of themcan be written in the form Γ =
5i=1 CiΓi for some constants C1, . . . ,C5. Thus, the adjoint action
can be viewed as (in fact is) a group of linear transformations of the vectors (C1, . . . ,C5). Theadjoint representation group is generated (via Lie equations) by the Lie algebraGA spanned by thesymmetries
∆i = cki jej ∂
∂ek, i = 1, . . . ,5,
where cki j are the structure constants in Table I. Explicitly, one has
∆1η(Γ) = C3∂η(Γ)∂C4
− 2C4∂η(Γ)∂C5
= 0, ∆2η(Γ) = C1∂η(Γ)∂C4
= 0, ∆3η(Γ) = 2C1∂η(Γ)∂C5
= 0.
(7)
The invariants η place a restriction on how far one can expect to simplify Γ by the action ofadjoint maps. For any other composition of adjoint maps there does not exist an α and β such thatsome of the coefficients c1,c2,c3,c4, and c5 in
X =5
i=1
ciΓi = Ad(exp(αΓi)) Ad(exp(βΓj))Γ i, j = 1, . . . ,5, (8)
are eliminated simultaneously.Based on (6)-(8) and the classification of the one-dimensional subalgebras of the GZEs Lie
algebra G, from Tables I and II one can obtain easily an optimal system of one-dimensional sub-algebras given by cΓ1 + Γ2,cΓ1 + Γ2 + Γ3,c1Γ1 + Γ2 + c4Γ4 + c5Γ5,c1Γ1 + Γ2 + c3Γ3 + c4Γ4 + c5Γ5,where ci =
CiC2, i = 1,2,3,4,5.
The Lie algebra, spanned by infinitesimal generators (5), is the direct sum of Γ2 and thenilpotent four-dimensional Lie algebra n4,1 (see Ref. 29).
B. Some symmetry reductions and exact solutions
In this subsection we use the whole Lie algebra and the optimal system of one-dimensionalsubalgebras calculated above to obtain symmetry reductions that yield solutions of (2) from solu-tions of corresponding systems of ordinary differential equations (ODEs). In order toobtain symmetry reductions and exact solutions, one has to solve the associated characteristicequations
dxξ1(x, t,u, v,F) =
dtξ2(x, t,u, v,F) =
duη1(x, t,u, v,F) =
dvη2(x, t,u, v,F) =
dFη3(x, t,u, v,F) . (9)
We consider the following cases.
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101501-5 E. Buhe and G. W. Bluman J. Math. Phys. 56, 101501 (2015)
Case 1. C3 = C4 = C5 = 0 and C1 = c, C2 = 1.Here, the solution of the corresponding characteristic system for the subalgebra cΓ1 + Γ2 a
combination of the space-time translation symmetries of the symmetry field Γ gives rise to travelingwave group-invariant solutions of the form
u = U(ξ), v = V (ξ), F = F(ξ), (10)
where ξ = x − ct is an invariant of the symmetry cΓ1 + Γ2, and c is an arbitrary constant. Thefunctions U, V , and F satisfy
2FU +U ′′ − 2βU(U2 + V 2) + cV ′ = 0,2FV − cU ′ − 2βV (U2 + V 2) + V ′′ = 0,(c2 − 1)F ′′ + 2(U2 + V 2)′′ = 0.
(11)
Integrating the third equation of (11) twice, we obtain
F =2(U2 + V 2)
1 − c2 + aξ + b, (12)
where a,b are two integration constants.Then, inserting (12) into the first and second equations of (11), we get
U ′′ + cV ′ = −2U(aξ + b),cU ′ − V ′′ = 2V (aξ + b), (13)
where c = ±
1 − 2β
. For (13), we consider two cases.(1) When ab , 0, we obtain the solution
U = [c1 cos(12
cξ) + c3 sin(12
cξ)]Ai(ζ) + [c2 cos(12
cξ) + c4 sin(12
cξ)]Bi(ζ),
V = [c1 sin(12
cξ) − c3 cos(12
cξ)]Ai(ζ) + [c2 sin(12
cξ) − c4 cos(12
cξ)]Bi(ζ),(14)
where c1,c2,c3,c4 are arbitrary constants, ζ = − 18
3
2a2 (8aξ + 8b + c2), and Ai,Bi are the Airy wave
functions of the first and second kind, respectively. Then, after substituting (14) into (12), we obtain
F = β[c1Ai(ζ) sin(cξ2) − c3Ai(ζ) cos(cξ
2) + c2Bi(ζ) sin(cξ
2) − c4Bi(ζ) sin(cξ
2)]2
+ [c1Ai(ζ) cos(cξ2) + c2Bi(ζ) cos(cξ
2) + sin(cξ
2) (c3Ai(ζ) + c4Bi(ζ))]2 + aξ + b.
(15)
The profile of solution (13)-(15) is given in Figs. 1 and 2.
FIG. 1. Evolution of travelling wave solution (14) and (15) with parameters a = 5,b = 80,c1= c = 1,c2= 2,c3= 3,c4= 4, β =10.
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101501-6 E. Buhe and G. W. Bluman J. Math. Phys. 56, 101501 (2015)
FIG. 2. Evolution of travelling wave solution (14) and (15) at x = 0 with parameters a = 5,b = 80,c1= c = 1,c2= 2,c3=
3,c4= 4, β = 50.
(2) When a = 0 and c2 + 8b > 0, we obtain the periodic solution
U = n1 cos(Aξ) + n2 sin(Aξ) + n3 cos(Bξ) + n4 sin(Bξ),V = −n1 sin(Aξ) + n2 cos(Aξ) − n3 sin(Bξ) + n4 cos(Bξ), (16)
where A = − 12 +
12
√c2 + 8b, B = − 1
2 −12
√c2 + 8b, and n1,n2,n3,n4 are arbitrary constants. Then,
after substituting (16) into (12), we get
F = β[n2 sin(Aξ) + n1 cos(Aξ) + n4 sin(Bξ) + n3 cos(Bξ)]2+[n1 sin(Aξ) − n2 cos(Aξ) + n3 sin(Bξ) − n4 cos(Bξ)]2 + b.
(17)
The profile of solution (16)-(17) is given in Figs. 3 and 4.Case 2. C3 = C4 = 0, C1 = c, and C2 = C5 = 1.The subalgebra cΓ1 + Γ2 + Γ5, which is a combination of the translation and rotation symme-
tries of the symmetry field Γ gives rise to the group-invariant solution of the form
G2(ξ) = u2 + v2,
u = G(ξ) cos(x + K(ξ)),v = G(ξ) sin(x + K(ξ)),F = F(ξ),
(18)
where ξ = x − ct is an invariant of the symmetry cΓ1 + Γ2 + Γ5, c is a constant wave speed, and thesimilarity functions G,K , and F satisfying the following similarity reduction equations:
FIG. 3. Evolution of travelling wave solution (16) and (17) with parameters b = 80,n1= 4,n2= 3,c3= 2,n4= c = 1, β = 10.
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101501-7 E. Buhe and G. W. Bluman J. Math. Phys. 56, 101501 (2015)
FIG. 4. Evolution of travelling wave solution (16) and (17) at x = 0 with parameters b = 80,n1= 4,n2= 3,c3= 2,n4= c =
1, β = 10.
G′′ + [(c − 2)K ′ + 2F − K ′2 − 1]G − 2βG3 = 0,
(2K ′ − c + 2)G′ + GK ′′ = 0,
(c2 − 1)F ′′ + 2GG′′ + 2G′2 = 0.
(19)
Three subcases arise as follows.Subcase I. c , 1.Integrating the second equation of (19) once after multiplication by G(ξ) and then the third
equation of (19) twice, we have
K ′(ξ) = c − 22+
λ1
2G2 ,
F(ξ) = 1c2 − 1
(−G2 + λ2 + λ3ξ),(20)
where λ1, λ2, and λ3 are integration constants.Substituting (20) into the first equation of (19), we get the nonlinear ODE
2( 11 − c2 − β)G3 + (c2
4− c + 2λ2 + 2λ3ξ)G + G′′ −
λ21
4G3 = 0. (21)
We are unable to solve (21). In the special case λ3 = 0, multiplying (21) by G′(ξ) and integratingonce, we obtain
4( 11 − c2 − β)G4 + [(c − 4)c + 8λ2]G2 + 4G′2 +
λ21
G2 = 8k1, (22)
where k1 is the integration constant. From (22), we find the general solution2√
c2 − 1G8(c2 − 1)k1G2 + c(c2 − 1)(c2 − 4c + 8λ2)G4 + 4[β(c2 − 1) + 1]G6 + (1 − c2)λ2
1
dG = ξ + k2,
(23)
where k2 is a second integration constant.
If c = ±
1 − 1β
, (23) becomes
dg
Ag2 + Bg + C= ξ + k2, (24)
where g = G2, A = ±
1 − 1β(1 − 1
β∓ 4
1 − 1β+ 8λ2),B = 8k1, and C = −λ2
1.
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101501-8 E. Buhe and G. W. Bluman J. Math. Phys. 56, 101501 (2015)
Solving the left side integral of (24), we get
ξ + k2 =
dg
Ag2 + Bg + C
=
1√
Aln | 2
A(Ag2 + Bg + C) + 2Ag + B |, A > 0,
1√
Aarcsinh( 2Ag + B
√4AC − B2
), A > 0,4AC > B2,
1√
Aln | 2Ag + B |, A > 0,B2 = 4AC,2Ag + B > 0,
− 1√
Aln | 2Ag + B |, A > 0,B2 = 4AC,2Ag + B < 0,
− 1√−A
arcsin( 2Ag + B√
4AC − B2),
A < 0,4AC < B2,
| 2Ag + B |< √B2 − 4AC.
.
(25)
Subcase II. c = 1.Here, system (19) becomes
G2F − K ′2 − K ′ − 1
+ G′′ − 2βG3 = 0,
G′ (2K ′ + 1) + GK ′′ = 0,G2′′ = 0,
(26)
which has the general solution
G =δ1ξ + δ2,
K =(δ2 + 2δ3) log (δ1ξ + δ2)
2δ1+ δ4 −
ξ
2,
F =8βδ3
1ξ3 + δ2
1
24βδ2ξ
2 + 3ξ2 + 1+ 6δ2δ1ξ (4βδ2 + 1) + 4
2βδ3
2 + δ22 + δ3δ2 + δ
23
8(δ1ξ + δ2)2,
(27)
where δ1, δ2, δ3, δ4 are arbitrary constants.Subcase III. K(ξ) = Aξ = (λ − 1)x + µt, i.e., A = λ − 1, Ac = −µ, (18) becomes
u = G(ξ) cos(λx + µt),v = G(ξ) sin(λx + µt),F = F(ξ).
(28)
Substitution of (28) into (2) results in the following system:
(c − 2λ) sin(η)G′ + cos(η)G(2F − µ − λ2) + cos(η)G′′ − 2β cos(η)G3 = 0,−(c − 2λ) cos(η)G′ + sin(η)G(2F − µ − λ2) + sin(η)G′′ − 2β sin(η)G3 = 0,(c2 − 1)F ′′ + 2GG′′ + 2G′2 = 0,
(29)
with η = λx + µt. Multiplying the first and second equations of (29) by cos(η) and sin(η), respec-tively, and then simplifying, we obtain the ODE
G(2F − µ − λ2) + G′′ − 2βG3 = 0. (30)
By integrating the third equation of (29) with respect to ξ twice, we obtain
(c2 − 1)F + G2 = λ1ξ + λ2, (31)
where λ1, λ2 are the resulting two integration constants.
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101501-9 E. Buhe and G. W. Bluman J. Math. Phys. 56, 101501 (2015)
In order to have a bounded solution, it follows that limξ→∞ F = 0, limξ→∞ F ′ = 0, limξ→∞G =0. Accordingly from Eq. (31), we set λ1 = λ2 = 0, and thus
F =G2
1 − c2 ,(32)
In view of (32), Eq. (30) becomes
2(β + 1c2 − 1
)G3 − G′′ +µ + λ2G = 0. (33)
In Refs. 20 and 21, the authors found a particular solution of (33) by applying the semi-inversemethod. Now, in order to obtain the general solution of (33), making an integrating factor G′, andintegrating once, Eq. (33) becomes
(β + 1c2 − 1
)G4 − G′2 +µ + λ2G2 = 2q, (34)
where q is integration constant. From (34), we get the general solutiondG
NG4 + MG2 − 2q= ξ, (35)
where N = β + 1c2−1
and M = µ + λ2. Solving the left hand integral of (35), we get the followingtwo results.
(1) When q , 0, we obtain
ξ =
dG
NG4 + MG2 − 2q=
√2
D1F( D1
2√
qG,D2), (36)
where D1 =
M2 − 8Nq − M , D2 =
12
M2+M
√M2−8Nq−4Nq
Nq, and F is the incomplete elliptic
integral of the first kind.(2) When q = 0, we obtain
ξ =
dG
NG4 + MG2 − 2q= ± 1√
Mln |
MN+
G2 + M
N
G| . (37)
Case 3. Ci, i = 1,2, . . . ,5 are arbitrary constants.For the most general generator Γ = c1Γ1 + Γ2 + c3Γ3 + c4Γ4 + c5Γ5, (ci = Ci
C2, i = 1,2,3,4,5), the
corresponding invariant solutions of (2) are given by
u(x, t) = −A1(ξ) sin(13
c3t3 + c4t2 − c5t) + A2(ξ) cos(13
c3t3 + c4t2 − c5t),
v(x, t) = A2(ξ) sin(13
c3t3 + c4t2 − c5t) + A1(ξ) cos(13
c3t3 + c4t2 − c5t),
F(x, t) = 12
c3t2 + c4t + H(ξ),
(38)
where A1(ξ), A2(ξ), and H(ξ) are functions of the similarity variable ξ = x − c1t that satisfy thenonlinear system of ODEs
−2βA32 + A2
−2βA2
1 + c5 + 2H+ c1A′1 + A′′2 = 0,
2βA31 − A1
−2βA2
1 + c5 + 2H+ c1A′2 − A′′1 = 0,
A21 + A2
2
′′+ (c2
1 − 1)H ′′ + c3 = 0.(39)
Integrating the third equation of (39) with respect to ξ twice, and letting c1 =
1 − 1β, (β <
0 or β ≥ 1), yields
H = β(A21 + A2
2) + c3βξ2 + κξ + µ, (40)
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101501-10 E. Buhe and G. W. Bluman J. Math. Phys. 56, 101501 (2015)
where κ, µ are integration constants. We substitute (40) into the first and second equations of (39) toobtain a linear ordinary differential system,
[c5 + 2(µ + κξ + c3βξ2)]A2 +
1 − 1
βA′1 + A′′2 = 0,
[c5 + 2(µ + κξ + c3βξ2)]A1 −
1 − 1
βA′2 + A′′1 = 0.
(41)
In order to solve (41), we consider the following subcases:Subcase I. c3 = 0, i.e., invariant solutions corresponding to the subalgebra Γ = c1Γ1 + Γ2 +
c4Γ4 + c5Γ5.(I-1) When κ = 0, we get the ODE system
(c5 + 2µ)A2 +
1 − 1
βA′1 + A′′2 = 0,
(c5 + 2µ)A1 −
1 − 1
βA′2 + A′′1 = 0,
(42)
whose general solution is given by
A1 = b1 cos(δξ) + b2 sin(δξ) + b3 cos(σξ) + b4 sin(σξ),A2 = −b1 sin(δξ) + b2 cos(δξ) − b3 sin(σξ) + b4 cos(σξ), (43)
with ∆ = β−1β+ 4(c5 + 2µ) > 0, δ = 1
2 (
β−1β+√∆), and σ = 1
2 (
β−1β−√∆), and bi, i = 1,2,3,4
are four arbitrary constants.Thus, from (38), (40), (43), and the transformation E = u + iv , we obtain the following exact
solution of the GZEs (1):
E(x, t) = [b1 cos(δξ) + b2 sin(δξ) + b3 cos(σξ) + b4 sin(σξ)][− sin (ζ) + i cos (ζ)]+[−b1 sin(δξ) + b2 cos(δξ) − b3 sin(σξ) + b4 cos(σξ)][cos (ζ) + i sin (ζ)],
F(x, t) = β[b1 cos(δξ) + b2 sin(δξ) + b3 cos(σξ) + b4 sin(σξ)]2+[−b1 sin(δξ) + b2 cos(δξ) − b3 sin(σξ) + b4 cos(σξ)]2 + c4t,
(44)
where ζ = c4t2 − c5t.(I-2) When β = 1, we get the ODE system
[c5 + 2(µ + κξ)]Ai + A′′i = 0, i = 1,2, (45)
whose solution is given by
A1 = A2 =√η[α1J1/3(2
3
√2κη3/2) + α2Y1/3(2
3
√2κη3/2)], (46)
with η = ξ + c5+2µ2κ ; J,Y are Bessel functions, and α1,α2 are two arbitrary constants.
Consequently, from (38), (40), (46), and the transformation E = u + iv , we obtain the exactsolution of the GZEs (1) given by
E(x, t) = √η[α1J1/3(23
√2κη3/2) + α2Y1/3(2
3
√2κη3/2)][(1 + i) cos (ζ) + (1 − i) sin (ζ)],
F(x, t) = 2η[α1J1/3(23
√2κη3/2) + α2Y1/3(2
3
√2κη3/2)]2 + κξ + µ + c4t,
(47)
where ζ = c4t2 − c5t.
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101501-11 E. Buhe and G. W. Bluman J. Math. Phys. 56, 101501 (2015)
Subcase II. c3 , 0, i.e., under the whole algebra Γ = c1Γ1 + Γ2 + c3Γ3 + c4Γ4 + c5Γ5.(II-1) When κ = 0 and c3 = −θ (θ > 0), we get the ODE system
[c5 + 2(µ − θ βξ2)]A2 +
1 − 1
βA′1 + A′′2 = 0,
[c5 + 2(µ − θ βξ2)]A1 −
1 − 1
βA′2 + A′′1 = 0,
(48)
whose general solution is given by
A1 =1√ξ[λ1Mk,m(φ) cos(ϕ) + λ2Mk,m(φ) sin(ϕ) + λ3Wk,m(φ) cos(ϕ) + λ4Wk,m(φ) sin(ϕ)],
A2 =1√ξ[−λ1Mk,m(φ) sin(ϕ) + λ2Mk,m(φ) cos(ϕ) − λ3Wk,m(φ) sin(ϕ) + λ4Wk,m(φ) cos(ϕ)],
(49)
where Mk,m(φ),Wk,m(φ) are the Whittaker functions, φ =
2βθξ2, ϕ = 12
1 − 1
βξ, k =
(β−1+8βµ+4βc5)√
232β3/2
√θ
, m = 14 , and λ1, λ2, λ3, λ4 are arbitrary constants.
Thus, from (38), (40), (49), and the transformation E = u + iv , we obtain the exact solution ofthe GZEs (1) given by
E(x, t) = [−A1 sin(ψ) + A2 cos(ψ)] + i[A2 sin(ψ) + A1 cos(ψ)],F(x, t) = β(A2
1 + A22) + c3βξ
2 + κξ + µ +12
c3t2 + c4t,(50)
where ψ = 13 c3t3 + c4t2 − c5t, and A1, A2 are given by (49).
(II-2) When β = 1 and c3 = −θ (θ > 0), we get an ODE system
[c5 + 2(µ + κξ − θξ2)]Ai + A′′i = 0, i = 1,2, (51)
which has the solution of the form
A1 = A2 = e−√
2ξ(θξ−κ)2√θ [µ1F(1
4−Ω, 1
2,ω) + µ2(2θξ − κ)F(3
4−Ω, 3
2,ω)], (52)
with Ω =√
2(2c5θ+4µθ+κ2)16θ3/2 ,ω =
√2(2θξ−κ)2
4θ3/2 , F is the generalized hypergeometric function, and µ1, µ2
are two arbitrary constants.Thus, from (38), (40), (52), and the transformation E = u + iv , we obtain the exact solution of
the GZEs (1) given by
E(x, t) = e−√
2ξ(θξ−κ)2√θ [µ1F(1
4−Ω, 1
2,ω) + µ2(2θξ − κ)F(3
4−Ω, 3
2,ω)]
×[(1 + i) cos (ψ) + (1 − i) sin (ψ)],F(x, t) = e
−√
2ξ(θξ−κ)√θ [µ1F(1
4−Ω, 1
2,ω) + µ2(2θξ − κ)F(3
4−Ω, 3
2,ω)]2
−θξ2 + κξ + µ − 12θt2 + c4t,
(53)
where ψ = − 13 θt3 + c4t2 − c5t.
III. CONSERVATION LAWS
Let x = (x1, x2, . . . , xn) be n independent variables and u = (u1,u2, . . . ,um) be m dependentvariables. Consider a system of r PDEs of kth-order given by
Pα[u] = Pα(x,u,u(1),u(2), . . . ,u(k)) = 0, α = 1,2, . . . ,r, (54)
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101501-12 E. Buhe and G. W. Bluman J. Math. Phys. 56, 101501 (2015)
where u(1) = uαi , u(2) = uα
i j, . . ., and uαi =
∂uα
∂xi,uα
i j =∂2uα
∂xi∂x j, . . .. We let U = (U1,U2, . . . ,UN)
denote arbitrary functions of the independent variables x and denote partial derivatives ∂/∂xi bysubscripts i, i.e., Uσ
i = ∂Uσ/∂xi,Uσi j = ∂
2Uσxix j
, etc.(1) The total derivative operators Di with respect to xi are
Di =∂
∂xi+ uα
i
∂
∂uαi
+ uαi j
∂
∂uαj
+ uαi jk
∂
∂uαjk
+ · · ·, (55)
where i, j, k, . . . = 1,2, . . . ,n and α = 1,2, . . . ,m.(2) Multipliers for PDE system (54) are a set of functions Λα[U] satisfying
Λα[U]Pα[U] = DiT i[U], (56)
for some functions T i[U].If Uσ = uσ(x) is a solution of PDE system (54), then from (56) we obtain the conservation law
DiT i[u] = 0 (57)
of system (54), and for each i, T i[u] is a flux.(3) The standard Euler operators with respect to the differentiable function U j and its deriva-
tives U ji ,U
ji1i2, . . . are the operators defined by
EU j =∂
∂U j− Di
∂
∂U ji
+ · · · + (−1)sDi1 · · · Dis
∂
∂U ji1· · ·is
+ · · ·, (58)
for each j = 1,2, . . . ,m.Λα[U] yields a set of multipliers for a conservation law of system (54) if and only if each
Euler operator (58) annihilates the left-hand side of (56), i.e.,
EU j(Λα[U]Pα[U]) ≡ 0, j = 1, . . . ,n, (59)
for arbitrary U,Ui,Ui j, . . ., etc.From determining Equation (59) for multipliers, we obtain the zeroth-order multipliers (with
the aid of GeM30) Λ1(x1, x2,U1,U2,U3), Λ2(x1, x2,U1,U2,U3) and Λ3(x1, x2,U1,U2,U3) for GZEs(2) which are given by
Λ1 = (1
2γ1t2 + γ2t + γ3)V,
Λ2 = −(1
2γ1t2 + γ2t + γ3)U,
Λ3 = − 1
12γ1t3 − 1
4γ2t2 +
112
t(−3γ1x2 + 12γ6x + 12γ4) − 14γ2x2 + γ7x + γ5,
(60)
where γi, i = 1,2, . . . ,7 are arbitrary constants and t = x1, x = x2,U = U1,V = U2.Then, from (56) and (60), we have the following seven conserved vectors of (2) satisfying
∂∂t
T ti [u, v,F] + ∂
∂xT xi [u, v,F] = 0, i = 1,2, . . . ,7 with
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101501-13 E. Buhe and G. W. Bluman J. Math. Phys. 56, 101501 (2015)
(T t1 ,T
x1 ) = [−( 1
12t3 +
14
t x2)Ft +14(t2 + x2)F − 1
4t2(u2 + v2),
( 112
t3 +14
t x2)Fx −12
t xF − (16
t3 +12
t x2)(uux + vvx) + 12
t x(u2 + v2)],
(T t2 ,T
x2 ) = [−1
4(t2 + x2)Ft +
12
tF − 12
t(u2 + v2),14(t2 + x2)Fx −
12
xF − 12(t2 + x2)(uux + vvx) + t(vux − uvx) + 1
2x(u2 + v2)],
(T t3 ,T
x3 ) = [−1
2(u2 + v2), vux − uvx],
(T t4 ,T
x4 ) = [tFt − F, t(2(uux + vvx) − Fx)],
(T t5 ,T
x5 ) = [Ft,2(uux + vvx) − Fx],
(T t6 ,T
x6 ) = [x(tFt − F), t(−xFx + F + 2x(uux + vvx) − u2 − v2)],
(T t7 ,T
x7 ) = [xFt,−xFx + F + 2x(uux + vvx) − u2 − v2].
(61)
Remark. One can show that there are no higher-order (≥1) multipliers for conservation laws ofEqs. (2) by applying the direct method. Since there are seven potential (nonlocal) variables resultingfrom conservation laws (61) of Eqs. (2), one could obtain a tree of up to 127 nonlocally related PDEsystems.9,31 Nonlocal symmetries of Eqs. (2) might arise from some of these nonlocally related PDEsystems through applying Lie’s algorithm to each nonlocally related system in the tree.9,31
IV. CONCLUSIONS
In this paper, we used Lie symmetry analysis to obtain reductions and new exact solutions ofthe generalized Zakharov equations through optimal systems of one-dimensional Lie subalgebras.The exact solutions contained some new solutions, including solutions involving Airy wave func-tions, Bessel functions, Whittaker functions, or generalized hypergeometric functions. The correct-ness of the solutions has been verified by substituting them back in to GZEs. Second, the conserva-tion laws of the GZEs have been constructed by using the direct (multiplier) method. Furthermorethese solutions can be used as benchmarks against numerical solutions. It is worth mentioning thatthe current study for GZEs can be further pursued from the point of view of Hamiltonian structuresand numerical simulations.
ACKNOWLEDGMENTS
The authors would like to express their sincere gratitude to the referees for many valuablecomments and suggestions which served to improve the article. The authors would like to thankAbdul Kara, Temuer Chaolu, Kamran Fakhar, and Gangwei Wang for helpful discussions andverifications of some of the calculations. This work is supported by the Research Project of ChinaScholarship Council (No. 201208155076), the Natural Science Foundation of Inner Mongolia (No.2013MS0118), the College Science Research Project of Inner Mongolia (No. NJZZ12182, No.NJZY13268), the program for Innovative Research Team of Hohhot University for Nationalities(No. CXTD1402), and the Natural Sciences and Engineering Research Council of Canada.
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