Similarity Reduction of Nonlinear Partial Differential Equations

In this work, the HB method is extended to search for similarity reduction of nonlinear partial differential equations. This method is generalized and will apply for a (2 + 1)-dimensional higher order Broer-Kaup System. Some new exact solutions of Broer-Kaup System are found.


Introduction
In the past few decades, there has been the noticeable progress in the construction of the exact solutions for nonlinear partial differential equations, which has long been a major concern for both mathematicians and physicists.The effort in finding exact solutions to nonlinear differential equation, when they exist, is very important for the understanding of most nonlinear physical phenomena.For instances, the nonlinear wave phenomena observed in fluid dynamics, plasma and optical fibers are often modelled by the bell shaped sech solutions and the kink shaped tanh solutions.
The systems (1)-(4) were given by Li et al [2] solving it via a transformation and tanh-function method to obtain many new exact solutions.Jain et al. [3] reduced a system to a simple (1 + 1)-dimensional nonlinear evolution equation through a simple transformation, and by using the new generally projective Riccati equation expansion method to explore many families of soliton-like and periodic solutions for it.Recently, Li et al. [4] have obtained some new types of multisoliton solutions for the systems (1)-( 4) by using some simple transformations as and homogenous balance method.
The homogenous balance (HB) method is a powerful tool to find solitary wave solutions of nonlinear partial differential equations.Fan et al. [5] presented an improved HB method to obtain more other kinds of exact solutions and introduced a continuation of [5] in [6].The traditional method for finding similarity reduction of nonlinear partial differential equations is to use classical Lie approach [7,8].However, the method involves tedious algebraic calculations and still can not be used to find all similarity solutions.Recently,Clarkson and Kruskal devloped a direct and simple method to find more similarity solutions of nonlinear PDEs.
In this work, the HB method is extended to search for similarity reduction of nonlinear partial differential equations.So, more solutions can be obtained by the improved HB method.This method is generalized and can be applied to other nonlinear partial differential equations [9][10][11][12][13][14][15].

Similarity Reduction of Nonlinear Partial Differential Equations
We describe the main steps of our method.For a given PDE, say in three variables, say , , x y t ( ) , , , , , we seek its similarity reductions in the form where α is a constant to determine by balancing between the highest order derivative of the linear terms of u and the nonlinear terms of u , where 0 , s u are regarded as undetermined functions.Substituting from Equation (6) into Equation ( 5) and collecting all terms of f with the same derivative and power.To make the associated equation be an ordinary equations of f and s , requiring ratios of their coefficients being functions of s , we obtain a set of determining equations for 0 , s u and other undermined functions, from which s and 0 u will be obtained.To explain this method, we will apply for a (2 + 1)-dimensional higher order Broer-Kaup system (1)-( 4), we suppose their similarity solutions are of the form

. s s x y t u u x y t v v x y t w w x y t p p x y t
are determined functions.Balancing the highest order of linear term with the nonlinear terms in every equations ( 1)-( 4) to determining 1 2 3 4 , , , , α α α α we obtain the Equation (64) take the following form ( ) ( ) Substituting Equations ( 9)-( 12) into the original system (1)-( 4) and collecting all terms of , , , f g h k with the same derivative and power leads to ) ( ) To make Equations ( 13)-( 16) be an ordinary differential equations of , , f g h and k only for s , the ratios of the coefficients of different derivative and power of f,g,h,k must be functions of s .That is to say, the following constrained conditions are satisfied ( ) There are freedoms in the determination of 0 0 0 0 , , , , u v w p s which can exploit the following rules, without loss of generality: Equations (91)and (94) into Equation (19), we get