New Jacobi Elliptic Function Solutions for the Generalized Nizhnik-Novikov-Veselov Equation∗

HONG BAO-JIAN

(Department of Basic Courses,Nanjing Institute of Technology,Nanjing,211167)

New Jacobi Elliptic Function Solutions for the Generalized Nizhnik-Novikov-Veselov Equation∗

HONG BAO-JIAN

(Department of Basic Courses,Nanjing Institute of Technology,Nanjing,211167)

In this paper,a new generalized Jacobi elliptic function expansion method based upon four new Jacobi elliptic functions is described and abundant solutions of new Jacobi elliptic functions for the generalized Nizhnik-Novikov-Veselov equations are obtained.It is shown that the new method is much more powerful in finding new exact solutions to various kinds of nonlinear evolution equations in mathematical physics.

generalized Jacobi elliptic function expansion method,Jacobi elliptic function solution,exact solution,generalized Nizhnik-Novikov-Veselov equation

1 Introduction

In recent years,due to the wide applications of soliton theory in natural science,searching for exact soliton solutions of nonlinear evolution equations plays an important and signi ficant role in the study on the dynamics of those phenomena(see[1]).Particularly,various powerful methods have been presented,such as inverse scattering transformation,Cole-Hopf transformation,Hirota bilinear method,homogeneous balance method,Backlund transformation,Darboux transformation,projective Riccati equations method and so on.In this paper,we discuss a generalized Nizhnik-Novikov-Veselov(GNNV)equation by our generalized Jacobi elliptic function expansion method(see[2])proposed recently.As a result,more new exact solutions are obtained.The character feature of our method is that,without much extra e ff ort,we can get series of exact solutions by using a uniform way.Another advantage of our method is that it also applies to general higher-dimensional nonlinear partial differential equations.

We consider the following GNNV equations(see[3–6]):

where a,b,c and d are arbitrary constants.For

the GNNV equations(1.1)are degenerated to the usual two-dimensional NNV equations (see[7–8]),which is an isotropic Lax extension of the classical(1+1)-dimensional shallow water-wave KdV model.When

we get the asymmetric NNV equation,which may be considered as a model for an incompressible fl uid.Some types of exact solutions of the GNNV equations have been studied in recent years(see[9–13]).

2 Summary of the New Generalized Jacobi Elliptic Functions Expansion Method

Given a partial differential equation with three variables x,y and t

we seek the following formal solutions of the given system by a new intermediate transformation:

where A0,Ai,Bi,Ci,Di(i=1,2,···,n)are constants to be determined later,ξ=ξ(x,y,t) is an arbitrary function with the variables x,y and t,the parameter n can be determined by balancing the highest order derivative terms with the nonlinear terms in(2.1),and E(ξ), F(ξ),G(ξ),H(ξ)are the arbitrary arrays of the four functions

respectively.The selection obeys the principle which makes the calculation more simple. We ansatz

where p,q,r,l are arbitrary constants which ensure denominator unequal to zero,so do the following situations.The four functions e,f,g,h satisfy the following restricted relations:

Substituting(2.4)along with(2.5a)–(2.5d)into(2.1),respectively,yields four families of polynomial equations for E(ξ),F(ξ),G(ξ),H(ξ).

Setting the coefficients of Fi(ξ)Ej1(ξ)j2G(ξ)j3H(ξ)j4(i=0,1,2,···;j1,j2,j3,j4=0,1; j1j2j3j4=0)to be zero yields a set of over-determined differential equations in A0,Ai,Bi, Ci,Di(i=1,2,···,n)and ξ(x,y,t).Solving the over-determined differential equations by Mathematica and Wu elimination,we obtain many exact solutions of(2.1)accroding to (2.2)and(2.3).

Obviously,if we choose the special values of p,q,r,l,m in(2.3),then we can get the results in[13–16],which has been discussed in[2].

3 Exact Solutions to the Generalized Nizhnik-Novikov-Veselov Equation

To seek the traveling wave solutions of(1.1),we make the gauge transformation

where k,τ,ω are constants to be determined later,and ξ0is an arbitrary constant.

Substituting(3.1)into(1.1)yields the ordinary differential equations(ODEs)of u(ξ), v(ξ),w(ξ)and integrating these ODEs makes the equations(1.1)to become

where C1and C2are integral constants.By balancing the highest-order of the linear term u′′and the nonlinear term u2in(3.2a),we obtain n=2.Thus we assume that(3.2a)has the following solutions:

satisfy(2.4)and(2.5a)–(2.5d).Substituting(2.4)and(2.5a)–(2.5d)along with(3.3)into (3.2a),respectively,and setting the coefficients of Fi(ξ)Ej1(ξ)j2G(ξ)j3H(ξ)j4(i=0,1,2,···; j1,j2,j3,j4=0,1;j1j2j3j4=0)to be zero yield an ODEs with respect to the unknowns ci(i=0,···,4),dj(j=1,···,10),ω,k,τ,p,q,r,l.After solving the ODEs by Mathematica and Wu elimination,we determine the following solutions:

Family 1: For p=0,we have

Case 1.

where k,τ,ξ0,c0,C1are arbitrary constants.ci(i=1,···,4)and dj(j=1,···,10)not mentioned here are zero,so do the following situations.

Therefore,from(2.3),(3.1),(3.3)and Case 1,we obtain the following solutions to the GNNV equations(1.1):

With the same process we derive the other three families of new exact solutions of(1.1), where

Remark 3.1Solutions u1,u6,u7,u8degenerate to solitary solutions when the modulus m → 1,and solutions u1,u3,u5,u7,u8degenerate to triangular function solutions when the modulus m→0.Here u6is just the solutions u1,u2,u3in[1].The other seven types of explicit solutions to(1.1)we obtained are not shown in the previous literature to our knowledge.

4 Conclusion

In this paper,we propose an approach for finding the new exact solutions for the nonlinear evolution equations by constructing the four new types of Jacobi elliptic functions(2.3). By using this method and computerized symbolic computation,we have found abundant new exact solutions of(1.1).More importantly,our method is much simple and powerful for finding new solutions to various kinds of nonlinear evolution equations.We believe that this method should play an important role in finding the exact solutions in mathematical physics.

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Communicated by Yin Jing-xue

35J20,35Q25

A

1674-5647(2012)01-0043-08

date:Sept.14,2009.

The Scienti fi c Research Foundation(QKJA2010011)of Nanjing Institute of Technology.