We study a two-dimensional integrable generalization of the Kaup-Kupershmidt equation, which arises in various problems in mathematical physics. Exact solutions are obtained using the Lie symmetry method in conjunction with the extended tanh method and the extended Jacobi elliptic function method. In addition to exact solutions we also present conservation laws which are derived using the multiplier approach.

The theory of nonlinear evolution equations (NLEEs) has made a substantial progress in the last few decades. Aspects of integrability of these equations have been studied in detail as it is evident from many research papers published in the literature. In many cases, exact solutions are required as numerical methods are not appropriate. Exact solutions of NLEEs arising in fluid dynamics, continuum mechanics, and general relativity are of considerable importance for the light they shed into extreme cases which are not susceptible to numerical treatments. However, finding exact solutions of NLEEs is a difficult task. In spite of this, many new methods have been developed recently that are being used to integrate the NLEEs. Among them are the inverse scattering transform [

In this paper we study a two-dimensional integrable generalization of the Kaup-Kupershmidt equation [

In the past few decades, the Lie symmetry method has proved to be a versatile tool for solving nonlinear problems described by the differential equations arising in mathematics, physics, and in many other scientific fields of study. For the theory and application of the Lie symmetry method see, for example, [

In the study of the solution process of differential equations (DEs), conservation laws play a central role. It is a well known fact that finding the conservation laws of DEs is often the first step towards finding the solution [

To study the two-dimensional integrable generalization of the Kaup-Kupershmidt equation (

The outline of the paper is as follows. In Section

The symmetry group of the system (

Applying the fifth prolongation, pr

One of the main reasons for calculating symmetries of a differential equation is to use them for obtaining symmetry reductions and finding exact solutions. This can be achieved with the use of Lie point symmetries admitted by (

Consider the first three translation symmetries and let

In this section we use the extended tanh function method which was introduced by Wazwaz [

In our case, the balancing procedure gives

Solving the resultant system of algebraic equations, with the aid of Mathematica, one possible set of values of

A profile of the solution (

Evolution of travelling wave solution (

In this subsection we obtain exact solutions of (

When

When

We now treat the above ODEs as our auxillary equations and apply the procedure of the previous subsection to system (

A profile of solutions (

Evolution of travelling wave solution (

Evolution of travelling wave solution (

In this section we construct conservation laws for

Consider a

We now construct conservation laws for the two-dimensional integrable generalization of the Kaup-Kupershmidt equation (

Due to the presence of the arbitrary functions,

In this paper we studied the two-dimensional generalization of the Kaup-Kupershmidt equation (

A. R. Adem and C. M. Khalique would like to thank the Organizing Committee of “Symmetries, Differential Equations, and Applications: Galois Bicentenary” (SDEA2012) Conference for their kind hospitality during the conference.