Fa-ren Wang and S Y Lou

School of Physical Science and Technology,Ningbo University,Ningbo,315211,China

Abstract To find symmetries,symmetry groups and group invariant solutions are fundamental and significant in nonlinear physics.In this paper,the finite point symmetry group of the combined KP3 and KP4 (CKP34) equation is found by means of a direct method.The related point symmetries can be obtained simply by taking the infinitesimal form of the finite point symmetry group.The point symmetries of the CKP34 equation constitute an infinite dimensional Kac-Moody–Virasoro algebra.The point symmetry invariant solutions of the CKP34 equation are obtained via the standard classical Lie point symmetry method.

Keywords: symmetry reduction,integrable system,Lie algebra

1.Introduction

Similarity solutions can be obtained in many ways,the most famous methods are the classical Lie group approach,the nonclassical Lie group approach and the direct method[1].To simplify the calculation,a simple direct method to derive symmetry reductions of a nonlinear system without involving any group theory was introduced by Clarkson and Kruskal(CK).This method has been applied to many nonlinear equations such as the Kadomtsev–Petviashvili equation [2],the Boussinesq equation [3],(2+1)-Dimensional General Nonintegrable KdV equation [4],Jimbo–Miwa equation [5]and the dispersive wave equations in shallow water [6].

The classical Lie group approach can be also important to find the Lie point symmetry groups of a nonlinear system[7].In [8],the symmetry groups of the KP equation via the traditional Lie group approach had been studied by David,Levi and Winternitz via the traditional Lie group approach.However,the method requires many algebraic operations and cannot obtain all the similar solutions,therefore,the direct method becomes a simple and effective method for us to find symmetry reduction.In [9],two different methods have been applied to the Whitham–Broer–Kaup equation,the results of the classical Lie group approach only got the form of Painlevé but the direct method got five different types of equations,therefore,the results of the classical Lie group approach is only a special case of the direct method.In recent years,similar methods have been applied to some new models,such as the Benney system [10] and Boundary-Layer equations[11].

Recently,a novel(2+1)-dimensional Korteweg–de Vries(KdV) extension,the combined KP3 (Kadomtsev–Petviashvili) and KP4 (CKP34) equation

is proposed by one of the present authors (Lou) [12].It has been proved that it has the Painlevé property,the B?cklund/Levi transformations and the residual nonlocal symmetry[12].It is worth mentioning that the CKP34 equation not only has the same properties as the KP3 equation and the KP4 equation but also has some different properties,such as soliton molecular solutions and D’Alembert solutions,which deserve to be discussed and studied.

In section 2 of this paper,the equivalence between the special form and general form of solutions is proved,meanwhile,we get the Lie point group of the CKP34 equation by a direct method.In section 3,we introduce point symmetry and the Kac-Moody–Virasoro symmetry algebra.In section 4,we apply the standard Lie point symmetry method to the CKP34 equation.Finally,section 5 is a summary and discussion.

2.Finite symmetry group of the CKP34 equation via a direct method.

The most general finite symmetry transformation of the CKP34 equation read

whereUandVsatisfy the same CKP34 equation

Substituting (3) into (1) and (2) and requiringU(ξ,η,τ)andV(ξ,η,τ)also can be the solutions of the CKP34 equation but with different independent variables (eliminatingUξτand its higher order derivatives by means of the CKP34 equation),we have

Obviously,WUandZUcannot be zero,and if ξx=0,there are no nontrivial solutions,so causing the coefficients ofandto vanish,the only possible case is

Equivalent to ξx,we know that ηy≠0 and τt≠0,next,we consider the coefficients of polynomials and get

under the above conditions,(7) and (8) can be further simplified to

then we have come to the conclusion that the form of

can be equivalent to the general form (3).

By substituting(10)into(1),we get a complex equation,so it is necessary to find some simple conditions,by combining (2) and (10),we get

Now we need to search for some special coefficients in the result of combining(1)and(10)by the above restrictions,our attention should be focused on the coefficients of polynomials read

according to (11),(12) and analyze different variables in equations,we get

assuming ξyis a function ofyandt,ξyis a function ofxandt,we will find contradictions on both sides of the equation in(11),so we know

under the above conditions,the coefficient ofUηandVξread

The polynomial coefficients in(13)related toUandVare zero and the results read

the constants δ andC1possess discrete values determined byin summary,the following theorem reads:

Theorem.IfU=U(x,y,t)andV=V(x,y,t)are thesolutions of the CKP34 equation,then so is{u,v}with(14)–(15).

According to the symmetry group theorem,we find that for the CKP34 equation,the symmetry group is divided into two sectors: the Lie point symmetry group which reads

and a coset of the Lie point group which is related to

the coset is equivalent to the reflected transformation ofy:y→-y.

If we denote bySthe Lie point symmetry group of the CKP34 equation,by σythe reflection ofy,Iis the identity transformation andC2≡{I,σy} is the discrete reflection group,the full Lie symmetry group GRCKPcan be expressed as

For the complex cKP3-4 equations,the symmetry group can be divided into six sectors read

The full symmetry groups,GCCKP,for the complex CKP34 equation are the product of the usual Lie point symmetry groupSand the discrete groupD3,

whereIis the identity transformation,σyis the reflection ofyand

andvcan be parallel tou.

3.Point symmetries and the Kac-Moody–Virasoro symmetry algebra

From the traditional method,we can simply take the arbitrary functionsx0,y0and τ in the forms

where ?is an infinitesimal parameter.By substituting (17)into (14) and (15) and using small parameter expansion method with respect to ?yields

Comparing (18),(19) and (20),we get three symmetry generators,

From (24),we know that the Kac-Moody algebra was constituted byK0andK1Λ,K2constitutes the Virasoro algebra.When we fix the arbitrary functions α,β and θ as special exponential functions or polynomial functionstmform=0,±1,±2,K,the generalized KMV algebra is reduced to the usual one.

Through the above discussion,we find the series symmetry approach can be equivalent to the Lie point symmetry method to the CKP34 equation and we can achieve three generators,which can be helpful for our following study.

4.Symmetry reductions of the CKP34 equation

In this section,we focus on the symmetry invariant solutions of the CKP34 equation related to the symmetries generated by

The symmetriesK0(α),K1(β) andK0(θ) establish a generalized KMV(Kac-Moody–Virasoro) algebra with the non-zero commutators

the above commutator [F,G] withF=(F1(u,v),F2(u,v))TandG=(G1(u,v),G2(u,v))T,here the superscriptTmeans the following three generators,

the transposition of the matrix,which is defined by

where α,β and θ are arbitrary functions oft.

Using the Lie point symmetriesK0(α),K1(β)andK0(θ)to the cKP3-4 equation(1),we can get two nontrivial symmetry reductions.

Reduction I:θ ≠0.For θ ≠0,we replace the arbitrary functions in the form:

with the new definitions (28),the group invariant condition becomes

because the process of solvingvcan be parallel tou,so we only solveuhere.Substituting (28) into (29) we get

here we give some equivalent transformations:

whereU(ξ,η)≡UandV(ξ,η)≡Vare invariant functions of the group with invariant variables ξ and η,combining (28),(29),(32)and vanishing the coefficients ofUξandUη,we get

so the first type of group invariant solutions become

By substituting (34) and (35) into (1) and (2),we can achieve the group invariant reduction equations about the group invariant functionsUandV,

it is worth mentioning that the reduction system (36) is Lax integrable with the fourth order spectral problem

On the other hand,in the reduction system (36),we can make the following transformation:

with respect to η,the system (36) can be equivalent to the Boussinesq equation.

Reduction II:θ=0,β ≠0.In this case,the group invariant condition becomes:

5.Summary and discussion

In this paper,a direct method was applied to find the finite point symmetry groups of the CKP34 equation,and whence the finite Lie point symmetry group is obtained,to find its related Lie symmetry algebra is quite straightforward.However,the simple ansatz (10) is not a universal formal for some other (2+1)-dimensional integrable systems.For example,in the Ablowitz–Kaup–Newell–Segur system,in order to find its generalized finite symmetry groups,the form of solutions should be modified to another form mentioned in [13],therefore,it is worthwhile for us to find the more general method.

Another method to find similarity reductions is the CK direct method.For the CKP34 equation,the results of the CK direct method can be equivalent to the results of(36)and this process can be divided into two parts;the one case can be proved thatUandVare not related to τ,and the other case is thatUandVare not related to η.We can substitute

into (1) and (2),then we can select a standardization coefficient which can be convenient for us to use the method mentioned in [1] to achieve the purpose of taking a special value and simplifying the equations and the standardization coefficient reads:

During our computational process,we found that the form of the CK direct method seems can be related to the B?cklund transformation of the CKP34 equation because both of them have the same coefficients and similar forms,which can be worthwhile for further research.

Recently,linear superposition in the general heavenly equation[14]had been found,the approximate method involves constructing Poisson brackets and using hodograph transformation.Whether this method can be applied to find symmetry reductions in different nonlinear systems is intriguing.

By using the Lie approach,we can reduce the KP equation to the Boussinesq equation [15],and then use the nonclassical symmetry reduction method to reduce the Boussinesq equation to the ordinary differential equation[16].Therefore,the CKP34 equation has many remaining problems for us to do further study.

Acknowledgments

The work was sponsored by the National Natural Science Foundations of China (Nos.11975131,11435005) and thank Professor M Jia for some helpful discussions in Ningbo University.

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