Rui and Qi Boundary Value Problems (2016) 2016:130 DOI 10.1186/s13661-016-0634-3
R E S E A R C H Open Access Bilinear approach to quasi-periodic wave solutions of the Kersten-Krasil’shchik coupled KdV-mKdV system
Wenjuan Rui1* and Xuemei Qi2
*Correspondence: [email protected] 1College of Science, China Abstract University of Mining and Technology, Xuzhou, 221116, The Hirato bilinear method is extended to construct quasi-periodic wave solutions for P.R. China the Kersten-Krasil’shchik coupled KdV-mKdV system. One- and two-periodic wave Full list of author information is solutions are obtained by means of a multidimensional Riemann theta function. The available at the end of the article asymptotic property of the quasi-periodic wave solutions is proved. It is shown that the quasi-periodic wave solutions reduce to the soliton solutions in an asymptotic small amplitude limit. PACS Codes: 02.30.Ik; 02.30.Gp; 05.45.Yv Keywords: Riemann theta function; soliton solutions; quasi-periodic wave solution
1 Introduction Nonlinear evolution equations (NLEEs) play important roles in many areas of science, and their study goes back a very long time. A variety of effective methods for constructing exact solutions of NLEEs have been developed, including the inverse scattering method, the Darboux transformation, the Hirota direct method, the algebro-geometrical approach [–], etc. Among the aforementioned methods, the Hirota method has proved particu- larly powerful. Once the corresponding bilinear form is derived, multi-soliton solutions for a nonlinear equation are usually obtained in a systematic way. Recently, many gen- eralizations of the Hirota method were developed. For example, a multiple exp-function method is proposed and provides an efficient way to an exact multiple wave solution pro- cedure that generalizes Hirota’s perturbation technique [, ]. The resulting multiple wave solutions contain resonant solitons and a linear combination of solutions of exponential waves, the latter of which presents an idea to construct linear subspaces of solutions for nonlinear equations. The Hirota bilinear method has been generalized to a broad class of nonlinear equations [] and the links with Bell polynomials were established []. It is interesting to note that the linear superposition principle can also apply to Hirota bilinear equations and present their resonant soliton solutions [], which is a special case of using the Riemann theta functions. Based on the Hitota bilinear method, Nakamura proposed an approach to quasi-periodic wave solutions of nonlinear equations [, ]. This method uses the multidimensional Riemann theta function directly and does not involve Lax pair representations and the complicated calculus on Riemann surfaces for the equations con-
© 2016 Rui and Qi. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro- vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Rui and Qi Boundary Value Problems (2016) 2016:130 Page 2 of 13
sidered. Recently, Fan, Hon, Ma, and Zhang have extended such method to investigate the breaking soliton equation, the discrete Toda lattice, the asymmetrical Nizhnik-Novikov- Veselov equation and the variable-coefficient mKdV equation, respectively [–]. In this paper, we would like to consider the following coupled KdV-mKdV system:
ut + uxxx –uux +vvxxx +vxvxx –uxv –uvvx =, (.) vt + vxxx –v vx –uvx –uxv =,
which was proposed as the classical part of one of superextensions of the KdV equation by Kersten and Krasil’shchik []. Taking v =,wehavetheKdVequation
ut + uxxx –uux =,
while for u = we get the mKdV equation
vt + vxxx –v vx =.
Therefore, system (.) can also be regarded as a type of coupling between the KdV and the mKdV equations. For system (.), Kersten and Krasil’shchik proved its complete in- tegrability by establishing the existence of an infinite series of symmetries and conserva- tion laws []; Kalkanli et al. presented its singular analysis and Lax pair by means of the Painlevé test and prolongation technique []; Hon and Fan devised an algebraic method to construct its solitary wave solutions and doubly periodic wave solutions []; Qin et al. applied the Bell polynomials to derive its bilinear form and N-soliton solutions []. How- ever, to the best of our knowledge, the quasi-periodic wave solutions for the system (.) have not been investigated. In the following, the Riemann theta function and the Hirato bilinear method will be applied to construct one- and two-periodic wave solutions for the system (.). In addition, a detailed asymptotic analysis procedure to the quasi-periodic wave solutions is presented and the relation between the quasi-periodic wave solutions and soliton solutions is established. This paper is organized as follows. In Section , we briefly introduce some main points on the Riemann theta function. In Section and Section , we construct one- and two- periodic wave solutions for the coupled KdV-mKdV system based on the Riemann theta function and bilinear method. Furthermore, the relation between the quasi-periodic wave solutions and soliton solutions is investigated. Finally, some concluding remarks are given in Section .
2 The bilinear form and the Riemann theta function To begin with, we introduce a useful bilinear form of system (.). Under the dependent variable transformation defined by []
∗ ∗ f u =–∂ ln ff , v = i ∂ ln ,(.) x x f Rui and Qi Boundary Value Problems (2016) 2016:130 Page 3 of 13
where f ∗ is the conjugation of f ,system(.) is transformed into the following bilinear form: · ∗ Dt + Dx f f =, (.) · ∗ DxDt + Dx f f =.
Following the Hirota bilinear method, the one-soliton solution for system (.)isofthe form
–ieη u =–∂ ln +eη , v = i ∂ ln ,(.) x x +ieη
where η = αx – αt + σ and α, σ are free constants. The two-soliton solution for system (.)isdenotedby η+η+A η η u =–∂x ln –e + e + e , (.) –ieη – ieη – eη+η+A v = i ∂x ln , +ieη + ieη – eη+η+A
with
ηj = αjx – αj t + σj, j =,, α – α eA = , α + α
and here αj, σj are constants. To construct quasi-periodic wave solutions of system (.), it is necessary for us to con- sider a generalized form of the bilinear equation (.). Assume the solution of (.)takes the form ∗ ∗ f u = u – ∂ ln ff , v = i ∂ ln ,(.) x x f
where u is a constant solution of (.). Submitting (.)into(.), we get · ∗ · ∗ H(Dx, Dt)f f = Dt + Dx –uDx f f =, (.) · ∗ · ∗ H(Dx, Dt)f f = DxDt + Dx –uDx + c f f =,
where c = c(t) is an integration constant. The constant c is vital in the construction of quasi-periodic solutions because elliptic functions do not usually satisfy equations with zero integration constants. In the following, we will construct quasi-periodic solutions for the bilinear equation (.) based on the multidimensional Riemann theta function [] ϑ(ξ, , s|τ)= e–π τ(n+s),n+s +πi ξ+,n+s ,(.) n∈ZN
n T N where · is the standard inner product of R , n =(n,...,nN ) ∈ Z ,wehavethecomplex T N T phase variable ξ =(ξ,...,ξN ) ∈ C , the complex parameter vector s =(s,...,sN ) , = Rui and Qi Boundary Value Problems (2016) 2016:130 Page 4 of 13
T (,...,N ) ,andthematrixτ =(τij) is a positive definite and real-valued symmetric N ×N matrix. In the definition of the Riemann theta function, the positive definiteness of τ guarantees (.) converges absolutely for all values of ξ. For simplicity, hereafter we use ϑ(ξ + , τ)= ϑ(ξ, ,|τ). Now let us see the periodicity of the Riemann theta function.
Definition Afunctiong(x, t)onCN × C is said to be quasi-periodic in t with funda-
mental periods T,...,Tk ∈ C if T,...,Tk are linearly dependent over Z and there exists a N k k function G(t, t) ∈ C × C such that for ∀(y,...,yk) ∈ C
G(x, y,...,yj + Tj,...,yk)=G(x, y,...,yj,...,yk), G(x, t,...,t,...,t)=g(x, t).
In particular, g(x, t) becomes periodic with T if and only if Tj = mjT.
Proposition [] Let ej be the jth column of N × NidentitymatrixIN , τj the jth column
of τ , and τjj the (j, j) entry of τ . Then the theta function ϑ(ξ, τ) has the periodic properties
ϑ(ξ + ej + iτj, τ)=exp(–πiξj + πτjj)ϑ(ξ, τ).
The vectors {ej, j =,...,N} and {iτj, j =,...,N} can be regarded as periods of the theta
function ϑ(ξ, τ) with multipliers and exp(–πiξj + πτjj), respectively.
In general, let ϑ(ξ, ,|τ)andϑ(ξ, ,|τ) be two Riemann theta functions, where = T T T (,...,N ) , =(,...,N ) ,andξ =(ξ,...,ξN ) , ξj = kjx+ljy+mjt +γj, j =,,...,N.For a polynomial operator H(Dx, Dy, Dt)withrespecttoDx, Dy,andDt, the following formula holds: H(Dx, Dy, Dt)ϑ ξ, ,|τ · ϑ(ξ, ,|τ) = C , , μ ϑ ξ, + , μ/|τ ,(.) μ
where C , , μ = H πin – μ/, k,πin – μ/, l,πin – μ/, m n∈ZN × exp –π τ(n – μ/), n – μ/ +πi n – μ/, – .(.)
From equations (.)and(.), we see that if C , , μ =, (.)
are satisfied for all possible combinations μ = ,;...,μN =,,thenϑ(ξ, ,|τ)and ϑ(ξ, ,|τ) are quasi-periodic solutions of the bilinear equation H(Dx, Dy, Dt)ϑ ξ, ,|τ · ϑ(ξ, ,|τ)=. Rui and Qi Boundary Value Problems (2016) 2016:130 Page 5 of 13
The coupled bilinear equation (.) is more difficult to deal with than a single bilinear equation due to the appearance of two equations. The following proposition plays an im- portant role in constructing its quasi-periodic wave solutions.
± Proposition [] Let C(μ) be given in (.), and make a choice such that j – j = , j =,,...,N. Then:
() If H(Dx, Dy, Dt) is an even function of the form
H(–Dx,–Dy,–Dt)=H(Dx, Dy, Dt), N then C(μ) vanishes automatically for the case when j= μj is an odd number, namely
N C(μ)|μ =, for μj = mod . (.) j=
() If H(Dx, Dy, Dt) is an odd function of the form
H(–Dx,–Dy,–Dt)=–H(Dx, Dy, Dt), N then C(μ) vanishes automatically for the case when j= μj is an even number, namely
N C(μ)|μ =, for μj = mod . (.) j=
3 One-periodic wave solutions and asymptotic properties In this section, we construct the one-periodic wave solutions for the system (.)with | ∗ | N = in Riemann theta function (.). Setting f = ϑ(ξ, , τ)andf = ϑ(ξ, , τ), f and f ∗ can be written as the following Fourier series in n:
f = ϑ ξ, , τ = ineπinξ–πn τ , n∈Z (.) ∗ f = ϑ ξ, , τ = (–i)neπinξ–πn τ , n∈Z
wherethephasevariableξ = kx + lt + γ , and the parameter τ >.
3.1 Construction of one-periodic wave solution In order to get the one-periodic wave solutions of the bilinear equation (.), we substitute (.)into(.), thus for i =,
∗ Hi(Dx, Dt)f · f μ μ = H πi n – , k ,πi n – , l i μ n∈Z Rui and Qi Boundary Value Problems (2016) 2016:130 Page 6 of 13
μ μ μ × exp –π τ n – , n – +πi n – , – μ × ϑ ξ,, τ μ = Ci(μ)ϑ ξ,, τ , μ =,, μ
where
μ μ μ C (μ)= πi n – l –π i n – k –πi n – u k n∈Z μ μ × exp –πτ n – + πi n – , μ μ μ C (μ)= –π n – kl +π n – k +π n – u k + c n∈Z μ μ × exp –πτ n – + πi n – .
According to Proposition , due to the fact H(Dx, Dt) is an odd function, we have C(μ =
) = . Meanwhile, H(Dx, Dt)isanevenfunctionandC(μ = ) = . Therefore, if the following equations are satisfied:
C (μ =)= πi n – l –π i n – k –πi n – u k n∈Z × exp –πτ n – + πi n – =, (.) C(μ =)= –π n kl +π n k +π n uk + c n∈Z × exp –πτn + πin =,
the Riemann theta functions (.) are exact solutions of equation (.). Taking ρ, ϑ, ϑ of the form
πτ ρ = e– , ϑ (ξ, ρ)=ϑ ξ, ,– τ = ρ(n–) exp πi n – ξ + , n∈Z (.) ϑ (ξ, ρ)=ϑ ξ, , τ = ρn exp πin ξ + , n∈Z Rui and Qi Boundary Value Problems (2016) 2016:130 Page 7 of 13
(a) (b)
Figure 1 One-periodic wave of u with parameters: k =0.1,γ =0,τ =2,u0 =0.(a)Perspective view of the wave. (b) Overhead view of the wave, with contour plot shown.
we transform equation (.) into a linear system about l and c
ϑl + ϑ k –uϑk =, (.) () ϑ kl + ϑ k –uϑ k + cϑ =,
where ϑ(p) (p) (p) d j ϑj = ϑj (, ρ)= , j =,;p =,,,,. dξ ξ=
Solving equation (.), we get
() ϑ ϑ ϑ – ϑ ϑ uϑ l =– k +uk, c = k + k .(.) ϑ ϑϑ ϑ
In this way, we get a one-periodic wave solution of system (.), ϑ(ξ, ,|τ) u = u – ∂ ln ϑ ξ, , τ ϑ ξ, , τ , v = i ∂ ln ,(.) x x | ϑ(ξ, , τ)
where ξ = kx + lt + γ , l is given by (.) and the other parameters, k, τ , γ ,andu,arefree. A one-periodic wave solution is determined by the two parameters k and τ .Figures and show one-periodic waves of u and v for one choice of the parameters, respectively. From Figures and , one can see that the one-periodic wave solution of system (.)is one dimensional and has two fundamental periods, and iτ ,inξ.Itcanbeviewedasa parallel superposition of overlapping one-solitary waves, placed one period apart.
3.2 Asymptotic property of one-periodic waves In the following, we will investigate the asymptotic properties of the one-periodic wave solution (.). The relation between the one-soliton solution (.) and the one-periodic wave solution (.) is given by the following theorem. Rui and Qi Boundary Value Problems (2016) 2016:130 Page 8 of 13
(a) (b)
Figure 2 One-periodic wave of v with parameters: k =0.1,γ =0,τ =2.(a)Perspective view of the wave. (b) Overhead view of the wave, with contour plot shown.
Theorem Suppose that (l, c)T is a solution of equation (.), let σ τ α u =, γ = + , k = ,(.) πi i πi
in which α and σ arethesameasthosein(.). Then the one-periodic solution (.) tends to the one-soliton solution (.), namely
u → u, v → v, as ρ → .
Proof We expand the coefficients of system (.) into power series of ρ
··· ··· ϑ =–πρ +πρ + , ϑ =π ρ +π ρ + , ··· ··· ϑ =–ρ + , ϑ =π ρ –π ρ + ,(.) () ··· ϑ =–π ρ +,π ρ + .
Assume that the solution of system (.) takes the form
l = l + lρ + lρ + ··· , (.) c = c + cρ + cρ + ··· ,
substituting equations (.)and(.)intoequation(.)andlettingρ → , we get
l =π k , c =. (.)
From (.)and(.), we have
c → , πil → π ki =–α,asρ → , ξ =πiξ – πτ = αx +πilt + σ (.) → αx – αt + σ = η,asρ → . Rui and Qi Boundary Value Problems (2016) 2016:130 Page 9 of 13
In order to verify that the one-periodic wave (.) tends to the one-soliton solution (.) when ρ → , we expand the function f in the form
f =+ρ ieπiξ – ie–πiξ – ρ eπiξ + e–πiξ + ··· .
It follows from (.)that
f =+ieξ – ρ eξ + ie–ξ + ρ eξ + e–ξ + ···
→ +ieη,asρ → .
Therefore,
∗ f → –ieη,asρ → .
we conclude that the one-periodic solution (.) tends to one-soliton solution (.)when ρ → .
4 Two-periodic wave solutions and asymptotic properties Let us consider two-periodic wave solutions for the coupled KdV-mKdV system (.). In the case of N =,wetakef and f ∗ as
f = ϑ(ξ, ,|τ) = e–π τn,n +πi ξ+,n n∈Z = (i)n+n e–π τn,n +πi ξ,n , n∈Z (.) ∗ f = ϑ ξ, ,|τ = e–π τn,n +πi ξ+ ,n , n∈Z = (–i)n+n e–π τn,n +πi ξ,n , n∈Z
T T T T where n =(n, n) ∈ Z , ξ =(ξ, ξ) ∈ C , ξj = kjx+ljt +γj, j =,,k =(k, k) , l =(l, l) , T T × =( , ) , =( , ) . τ is a positive definite and real-valued symmetric matrix, which can be written as
τ =(τpq)×, τ >,τ >,ττ – τ >.
4.1 Construction of two-periodic waves
By means of Proposition , noting that the fact that H(Dx, Dt) is an odd function, its cor- responding constraint equations in (.) vanish automatically for μ = (, ), (, ). Simi-
larly, the constraint equations of H(Dx, Dt) vanish for μ = (, ), (, ). Therefore, the theta Rui and Qi Boundary Value Problems (2016) 2016:130 Page 10 of 13
functions (.) are solutions of the bilinear equation (.) if the following equations:
μ μ μ πi n – , l –π i n – , k –u πi n – , k n∈Z μ μ μ × exp –π τ n – , n – +πi n – , – =, μ=(,),(,) μ μ μ –π n – , k n – , l +π n – , k (.) n∈Z μ +u π n – , k + c μ μ μ × exp –π τ n – , n – +πi n – , – =, μ=(,),(,)
are satisfied. Taking
πτ – pq ρpq = e , p, q =,,
ρ =(ρ, ρ, ρ), s ϑ (ξ, ρ)=ϑ ξ, – ,– r τ r s s s = exp –π τ n – r , n – r +πi ξ + – , n – r . (.) n∈Z – s (n –s )(n –s ) = exp πi ξ + , n – r ρ p r,p q r,q , pq n∈Z p,q=
sr =(sr, sr), r =,,,,
s = (, ), s = (, ), s = (, ), s = (, ),
equation (.) can be transformed into a linear system about l, l, c,andu,
(l ·∇)ϑr +(k ·∇) ϑr –u(k ·∇)ϑr =, r =,, (ξ,ξ)=(,) (.) (k ·∇)(l ·∇)ϑr +(k ·∇) ϑr –u(k ·∇) ϑr + cϑr =, r =,, (ξ,ξ)=(,)
where ∇ =(∂ξ, ∂ξ), k ·∇= k ∂ξ + k ∂ξ. Solving equation (.), we can get a two- periodic wave solution of the system (.)
∗ ∗ f u = u – ∂ ln ff , v = i ∂ ln ,(.) x x f
∗ where f , f ,andl, l, c, u are given by equations (.)and(.), and the other parameters,
k, k, τ, τ, τ, are free. A two-periodic wave solution is specified by the four parameters
k, k, τ,andτ. Rui and Qi Boundary Value Problems (2016) 2016:130 Page 11 of 13
4.2 Asymptotic property of two-periodic waves SimilartoTheorem, the relation between the two-soliton solution (.)andthetwo- periodic solution (.) can be given as follows.
T Theorem Suppose that (l, l, u, c) is a solution of system (.), let
σj τjj αj γ = + , k = , j =,, j πi i j πi (.) A τ = , π
where σj, αj, j =,,and A are given in (.). Then the two-periodic solution (.) tends to the two-soliton solution (.), that is,
u → u, v → v, as ρ, ρ → .
Proof We expand the periodic function f in the following form: f =+i eπiξ – e–πiξ e–πτ + i eπiξ – e–πiξ e–πτ , – eπi(ξ+ξ) + e–πi(ξ+ξ) e–π(τ+τ+τ) + ··· .
It follows from (.)that
ξ ξ ξ+ξ–πτ –ξ –ξ f =+ie + ie – e – iρe – iρe –ξ–ξ–πτ ··· – ρρe + ξ ξ ξ ξ → +ie + ie – e + +A , ∗ ξ ξ ξ +ξ +A f → –ie – ie – e ,asρ, ρ → ,
in which ξj =πiξj – πτjj = αjx +πiljt + σj, j = , . Thus the two-periodic solution equa- tion (.) tends to the two-soliton solution equation (.) provided that
→ → → c , πilj –αj , j =,, asρ, ρ .
Assume the solution of system (.) is in the following form: ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ () () () l l l l ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ () ⎟ ⎜ () ⎟ ⎜ () ⎟ ⎜l ⎟ ⎜l ⎟ ⎜l ⎟ ⎜l ⎟ ⎜ ⎟ = ⎜ ⎟ + ⎜ ⎟ ρ + ⎜ ⎟ ρ + o(ρ , ρ ). (.) ⎝ ⎠ ⎜ () ⎟ ⎜ () ⎟ ⎜ () ⎟ u ⎝u ⎠ ⎝u ⎠ ⎝u ⎠ c c() c() c()
Expanding the coefficient of system (.) and substituting (.)into(.), together with () taking u =,wethenobtain
() () → c =, πilj +(πi) kj =, j =,, asρ, ρ . Rui and Qi Boundary Value Problems (2016) 2016:130 Page 12 of 13
Thus we have
→ → → → u , c , πilj π kj i =–αj , j =,, asρ, ρ .
Therefore we conclude that the two-periodic solution (.) tends to the two-soliton solu-
tion (.)asρ, ρ → .
5Conclusions In this paper, the Kersten-Krasil’shchik coupled KdV-mKdV system is investigated. Based on the Hirato bilinear method and the Riemann theta function, the one-periodic and two- periodic wave solutions for the system are constructed. In addition, the asymptotic prop- erties of the quasi-periodic wave solutions are proved. It is shown that the quasi-periodic wave solutions converge to the soliton solutions in a small amplitude limit. It is natural to ask whether we can get multiperiodic wave solutions in the case of N >.However, the solvability of system (.) is the key to the construction of multiperiodic wave solu- tions. As the number of unknown parameters is less than the number of equations, we cannot get multiperiodic wave solutions directly when N > . How to get multiperiodic wave solutions when N > ? Such a question will be investigated in the future.
Competing interests The authors declare that no conflict of interest exists in the submission of this manuscript.
Authors’ contributions The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.
Author details 1College of Science, China University of Mining and Technology, Xuzhou, 221116, P.R. China. 2School of Resources and Geosciences, China University of Mining and Technology, Xuzhou, 221116, P.R. China.
Acknowledgements This work was supported by the Fundamental Research Funds for the Central Universities (no. 2014QNB50).
Received: 7 May 2016 Accepted: 27 June 2016
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