Ya-Rong Xia(夏亞榮),Xiang-Peng Xin(辛祥鵬), and Shun-Li Zhang(張順利)Center for Nonlinear Studies,School of Mathematics,Northwest University,Xi’an 710069,China

2School of Information Engineering,Xi’an University,Xi’an 710065,China

3School of Mathematical Sciences,Liaocheng University,Liaocheng 252059,China

1 Introduction

Conservation laws,essential in the study of differential equations mathematically and physically,propose one of the primary principles to formulate and investigate models,especially in existence,uniqueness and stability of solutions.In addition,the integrability of the system is quite possible should conservation laws exist in it.[1?2]For conservation laws,different methods have been mobilized.The celebrated Noether’s theorem[3]proves to be a systematic and efficient approach in finding conservation laws of PDEs unless there exists a Lagrangian.However,there exist some equations not having a Lagrangian.Hence the Noether’s theorem cannot be used to obtain conservation laws directly because of the equation symmetries.This,however,can be solved with the general concept of nonlinear self-adjointness proposed by Ibragimov,[4?7]and Gandarias to construct the conservation laws for any differential equation.[8]This procedure can be true of classes of single differential equations of any order but of the systems where the number of equations is equal to that of dependent variables.[9?11]

On the other hand,it is an important and major subject to seek exact solutions and interactions among solutions to nonlinear equation to explain some physical phenomena further.The special solutions to an integrable system can be derived from many e ff ective methods such as symmetry reductions,[12]the variable separation approach,[13]the inverse scattering transformation approach,[14]the Darboux transformation(DT),[15?16]thetransformation(BT),[17]the bilinear method,[18]and Painlev′e analysis,[19]to name just a few.However,it is difficult to find the interaction solutions among different types of nonlinear excitations besides the soliton-soliton interaction.Recently,Lou etal.made a breakthrough in interaction solutions between solitons and any other types of nonlinear soliton waves by using two equivalent simple methods:the truncated Painlev′e analysis and the generalized tanh expansion approaches,[20?21]which are proved to be e ff ective for more types of solutions to many integrable systems.

This paper concentrates on investigating the nonlinear self-adjointness,conservation laws and interaction solutions between a soliton and cnoidal wave[22?26]of the(2+1)-dimensional modi fi ed dispersive water-wave(MDWW)system,which can be written as

system(1),modeling nonlinear and dispersive long gravity waves in two horizontal directions on shallow waters of uniform depth.MDWW is derived from the famous Kadomtsev–Petviashvili(KP)equation with the symmetry constraints.[27]In Refs.[28]–[29],Painlev′e–B¨acklund transformations,along with a multilinear variable separation approach help a lot in securing abundant propagating localized excitations.Reference[30]shows many new types of non-traveling solutions acquired via a further generalized projective Riccati equation method.In[31],the extended mapping approach assists in getting some nonpropagating and propagating solitons.Reference[32]en-gages in new types of interactions between solitons such as a compacton-like semi-foldon and a compacton,a peakonlike semi-foldon and a peakon based on new variable separation solutions with arbitrary functions for MDWW(1)by using the projective Riccati equation expansion.In Ref.[33],special types of periodic folded waves are derived from the WTC truncation method.In Ref.[34],Hirota bilinear method is of great assistance in constructing multiple soliton solutions with arbitrary functions for system(1).For system(1),Ref.[35]emphasizes symmetry reduction.However,the research into the nonlinear self-adjoint,conservation law and soliton-cnodial wave solution of Eqs.(1)have not been mentioned in the above literature.

This paper is arranged as follows.Section 2 introduces the main notations and theorems used in this paper.In Sec.3,the nonlinear self-adjointness for the(2+1)-dimensional(MDWW)system will be discussed,which is a vital link in applying Ibragimov’s theorem.In Sec.4,based on Lie symmetry analysis acquired and Ibragimov’s theorem,conservation laws of system(1)are established.In Sec.5,we derive new explicit interactions solutions between solitons and cnoidal periodic waves by the truncated Painlev′e analysis and the consistent tanh expansion(CTE)method for the(2+1)-dimensional MDWW system.In the last section,some conclusions and discussion will be given.

2 Preliminaries

This section aims to present the notations and theorems used in this paper.

Definition 1(Ref.[6])Consider a system of equations

with n independent variables x=(x1,...,xn),m dependent variables u=(u1,...,um)and where u(s)denotes the set of the partial derivatives of s-th order of u.The adjoint equation to Eqs.(2)is

with

where L is the formal Lagrangian for Eq.(2)given by

with v=(v1,...,vm)as new dependent variables,vα=vα(x),and δ/δuαas the variational derivative

Definition 2(Ref.[7])The system(2)is said to be nonlinearly self-adjoint if the following equations hold:

with ?(x,u)/=0,whereare undetermined coefficients,and ? is the m-dimensional vector ? =(?1,...,?m).

In Ref.[6],Eqs.(3)succeeds the symmetries of the system(2),which has been proved by Ibragimov.In other words,if the system(2)admits a point transformation group with a generator

then the adjoint system(3)admits the operator(7)extended to the variables vαby the formula

Theorem 1(Ref.[6])Any in fi nitesimal symmetry(Lie point,Lie B¨acklund,nonlocal)

of a system equations(2)provides a conservation law Di(Ci)=0 for the system of differential equations consisting of Eqs.(2)and the adjoint Eqs.(3).The conserved vector is given by

and

3 Nonlinear Self-Adjointness of System(1)

For system(1),according to Definition 1,the following formal Lagrangian can be deduced

whereandare two new dependent variables.The adjoint system of the system(1)is

where,in this case

with Dx,Dyand Dtdenoting the operator of total differentiation with x,y,and t respectively.Should Eq.(10)be considered,the adjoint system(11)for system(1)will change into

System(1)is not recovered if u is substituted forand v for,so system(1)is not self adjoint.[10]Based on Definition 2,nonlinearly self-adjoint will the system(1)become if each equation(i=1,2)of the adjoint system(12)satis fi es the following condition

with regular undetermined coefficients λij(i,j=1,2)after substituting the following expression

with ?(x,y,t,u,v)/=0 or ψ(x,y,t,u,v)/=0.Were the differential consequences of(14)to be introduced,system(12)split into the following equations for the coefficients λij(i,j=1,2)

and into the system for the substitution(14)

Once they are solved,the following solution will come

where g1,g2,g3are arbitrary functions of t,and g4of y,and the dot over the function denotes its derivative with respect to its variable.Then,according to the Definition 2,system(1)is nonlinearly self adjoint.

4 Lie Symmetries and Conservation Laws of System(1)

The performance of corresponding Lie symmetry analysis by classical lie group method is the prerequisite to derive conservation laws for system(1).It needs to consider a one-parameter Lie group of in fi nitesimal transformations

with a small parameter ?? 1.The vector field related to the above transformations can be described as

Then the invariance of system(1)under transformation(17)makes the functions ξ1,ξ2,ξ3,η1,η2take the form

where f1is arbitrary function of y,f2,f3of t,and the dot over the functions means their derivative with respect to their variable.An in fi nite-dimensional Lie algebra of symmetries is resulted from the existence of the arbitrary functions.A general element of this algebra is depicted as

where

What follows is to apply the Theorem 1 to seek for conservation laws of system(1).For(1),the adjoint equation is given by

and the Lagrangian in the symmetrized form

Consider Theorem 1,the corresponding vector fields can be written as

The conservation law is decided by

Here the conserved vector C=(C1,C2,C3)is given by(9)and the concrete forms are as follows

Substituting(25)into(28),it will change into

with

In regard to(21),we consider the following cases.

Case 1For the generator

the Lie characteristic functions are

one can obtain the conservation vector of(1)

Case 2For the generator

the Lie characteristic functions are

we can get the conservation vector of(1)

Case 3For the generator

the Lie characteristic functions are

we derive the conservation vector of(1)

Remark 1Clearly,the above conservation vector Ci(i=1,2,3)includes an arbitrary solutionto adjoint Eqs.(24),so the number of the conservation laws it presents is in fi nite.

5 Soliton-Cnoidal Wave Interaction Solutions of System(1)

Obviously,the Painlev′e analysis is one of the e ff ective approaches for special solutions to nonlinear physical systems.For the(2+1)-dimensional MDWW system,its truncated Painlev′e expansion can be expressed as

with u0,u1,v0,v1,v2,? being the functions of x,y and t.By substituting Eq.(29)into system(1)and vanishing all the coefficients of different powers of 1/? comes

and then we obtain

which is the solution to the MDWW system,and the field

? satis fi es the following Schwarzian form

where λ is an arbitrary integral parameter,and

The Schwarzian form(32)is invariant under the M¨obious transformation

That is to say,Eq.(32)bears three symmetries σ?=d1,σ?=d2?,and σ?=d3?2with arbitrary constants d1,d2and d3.

Adopting the following straightening transformation,

where w is the function of x,y,and t.After substituting the expression(33)into system(31),the equivalent solutions to MDWW system come as

and the equivalent compatibility condition for w as

where

Clearly,the solutions(34)are derived from the transformation(33),where the usual truncated Painlev′e expansion approach is converted into the most general extension of the special tanh function expansion method,so it can be said the solutions(34)are the generalization of the usual tanh function expansion method.Here we can obtain the solution(34)by the CTE approaches.[36]

For the MDWW system(1),the application of leading order analysis can result in the following generalized truncated tanh function expansion

where u0,u1,v0,v1,v2and w are functions of x,y,and t.Substituting expression(36)into system(1)and vanishing all the coefficients of tanhi(w),we have

and then we deduce the same solution(34)to the MDWW system(1)with the consistent condition(35).

Fig.1 The soliton-cnodial periodic wave solution to u:(a)The pro fi le of the special structure with t=0 and y=0.(b)The pro fi le of the special structure at t=0 and x=0.(c)Perspective view of the wave.

Fig.2 The soliton-cnodial periodic wave solutions to v:(a)The pro fi le of the special structure with t=0 and y=0.(b)The pro fi le of the special structure with t=0 and x=0.(c)Perspective view of the wave.

The above shows that the single soliton(or solitary wave)solution to the MDWW system(1)is only a straightened solution w=k1x+l1y+d1t to Eq.(35),which implies that to find the interaction solutions between solitons and other nonlinear excitations,what is needed is to acquire the solution to Eq.(35).In this paper we focus on the following special Jacobi elliptic function

as the solution to Eq.(35),which characterizes the interactions between a soliton and a cnoidal wave.h1,h2,h3,q1,q2,q3,λ,m and n are determined later.In(38),sn(z,m)is the usual Jacobi elliptic sine function and

is the third type of incomplete elliptic integral.By substituting(38)into(35)and solving the over-determined equations with the help of maple will come

where h2,h3,λ,m,n,q1,q2and q3are arbitrary constants.Substituting Eqs.(37),(38),and(39)into(36),we can obtain the interaction solution between soliton and cnoidal periodic waves.The result is omitted here because of its prolixity.Corresponding images are as follows and the parameters used in the fi gure are selected as{h2=1.4,h3= ?0.5,λ= ?0.3,q1= ?0.9,q2= ?0.5,q3=0.2,m=0.8,n=0.5}.

Remark 3Figures 1 and 2 illustrate the soliton-cnoidal periodic wave solutions to the fields u and v describing a soliton travels on a cnoidal wave background for the MDWW system.Clearly,the interaction between the soliton and every peak of the cnoidal periodic wave is elastic as phase changes.Solutions and fi gures obtained in this paper might be helpful in further understanding the propagation of nonlinear and dispersive long gravity waves on shallow waters.

6 Summary and Discussion

It is proved that the(2+1)-dimensional MDWW system(1)is nonlinearly self-adjoint.With the support of the general theorem of conservation laws by Ibragimov,[6]the property can be applied to construct countless conservation laws for(1).Mathematically,the basic conserved quantity can be applied in obtaining various estimates for smooth solutions and de fi ning suitable norms for weak solutions,so it is worthy to be further investigated.

In addition,with the truncated Painlev′e analysis and the CTE method,the soliton-cnoidal wave solution to system(1)is obtained.A good understanding of the solutions to system(1)is very helpful for coastal and civil engineers in applying the nonlinear water model to coastal harbor design.For their practicability,the study on the CTE method and more types of the interaction solutions among different kinds of nonlinear excitations should be furthered.

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