Some exact solutions to the generalized Korteweg – deVries equation and the system of shallow water wave equations

Abstract. In this paper, we establish exact solutions for nonlinear evolution equations in mathematical physics. The exp-transform method is proposed to seek solitary solutions, periodic solutions and compaction-like solutions of nonlinear differential equations. The generalized KdV equation and the system of the shallow water wave equation are chosen to illustrate the effectiveness and convenience of the method.


Introduction
Many phenomena in physics and other fields are often described by non-linear partial differential equations (PDE's), such as fluid dynamics, plasma physics, mathematical biology, solid-state physics and chemical kinetics.When we want to understand the physical mechanism of natural phenomenon described by non-linear PDE, exact solutions for the nonlinear PDE have to be explored, thus the methods for deriving exact solutions for the governing equations have to be developed.Exploring exact solutions of nonlinear PDE's has become one of most important topics in mathematical physics.
The rest of the paper is organized as follows.In Section 2, we describe the exp-transform method for finding travelling wave solutions of nonlinear evolution equations and c Vilnius University, 2013 give the main steps of the method here.In the subsequent sections, in Section 3 and Section 4, we illustrate the method in detail with the generalized KdV equation and the system of the shallow water wave equations.Finally, conclusions and discussion are given.

Exp-transform method
We consider a general nonlinear PDE in the form where λ is constant, we can rewrite Eq. ( 1) in the following nonlinear ODE: The exp-transform method is based on the assumption that travelling wave solutions can be expressed in the following form [17][18][19][20][21][22][23][24]: where a i , b i are unknown constants.To determined values of m and n, we balance the linear term of highest order in Eq. ( 2) with the highest order nonlinear term.
ω(η) is a solution of the following first-order ordinary differential equations containing exponential functions (SET means set of exact solutions) in various combinations: where cd > 0. Substituting (3) into Eq.( 2) along with their derivations relating to the given ODEs A to C and yields a set of algebraic equations for e iω(η) .Setting the coefficients of e iω(η) to zero yields a set of over-determined algebraic equations with respect to the parameters c, d, a i , b i , λ 3 The generalized Korteweg-de Vries equation (GKdV) We next examine the GKdV equation: The wave variable u(x, t) = u(η), where η = x − λt, carries the GKdV equation (4) into a system of ODEs Case 1.We suppose that the solution of Eq. ( 5) can be expressed as where ω (η) = ce ω + ade −ω .Eq. ( 6) can be re-written in an alternative form as follows: to determined values of m and n, we balance the linear term of highest order in Eq. ( 5) with the highest order nonlinear term.By simple calculation, we have It is easy to find that n = m + 2 by balancing u 2 u with u (5) .Solutions for m = 0 and ω (η) = ce ω + ade −ω : Substituting ( 7) into ( 5) and collecting the coefficients of e iω(η) we obtain a system of algebraic equations for a i and λ.By solving this system, we obtain the following solutions: Solutions for m = 1 and ω (η) = ce ω + ade −ω : Substituting ( 8) into ( 5) and collecting the coefficients of e iω(η) we obtain a system of algebraic equations for a i , b j and λ.By solving this system, we obtain the following solutions: Case 2. We suppose that the solution of Eq. ( 5) can be expressed as where ω (η) = a √ ad 2 e 2ω(η) − acd.By balancing u 2 u with u (5) : Solution for m = 0 and ω (η) = a √ ad 2 e 2ω(η) − acd: Substituting ( 9) into (5), and collecting the coefficients of e iω(η) we obtain a system of algebraic equations for a i and λ.By solving this system, we obtain the following solutions: Solutions for m = 1 and ω (η) = a √ ad 2 e 2ω(η) − acd: Substituting ( 10) into ( 5) and collecting the coefficients of e iω(η) we obtain a system of algebraic equations for a i , b i and λ.By solving this system, we obtain the following solutions: Case 3. We suppose that the solution of Eq. ( 5) can be expressed as where ω (η) = √ ad 2 e −2ω(η) − acd.It is easy to find that n = m + 2 by balancing u 2 u with u (5) .
Nonlinear Anal.Model.Control, 2013, Vol. 18, No. 1, 27-36 4 The system of the shallow water wave equation We first consider the system of the shallow water wave equation in order to demonstrate the exponential transform method The wave variable η = x − λt carries Eq. ( 12) into the ODE where by integrating once the second equation and the constant integrations is zero, we find Substituting Eq. ( 14) into the first equation of Eq. ( 13) we obtain Case 1.We suppose that the solution of Eq. ( 15) can be expressed as where ω (η) = ce ω + ade −ω .Eq. ( 16) can be re-written in an alternative form as follows: To determine values of m and n, we balance the linear term of higest order in Eq. ( 15) with the highest order nonlinear term.By simple calculation, we have Solutions for m = 0 and ω (η) = ce ω + ade −ω : Substituting ( 17) into (15) and collecting the coefficients of e iω(η) we obtain a system of algebraic equations for a i and λ.By solving this system, we obtain the following solutions: Solutions for m = 1 and ω (η) = ce ω + ade −ω : Substituting ( 18) into ( 15) and collecting the coefficients of e iω(η) we obtain a system of algebraic equations for a i , b i and λ.By solving this system, we obtain the following solutions: , where b 1 = 0, where λ = 8cdb 0 /(a 0 − 2db 1 ), b 0 = 0, M = 2 + √ 3 + 3 + 2 √ 3 and N = 1 + √ 3.
Case 2. We suppose that the solution of Eq. ( 15) can be expressed as solutions include hyperbolic function solution, trigonometric solution and exponential function solution.It was formally derived that the solutions leads to both solitary solutions and periodic solutions for the first case.The performance of the exp-transform method is reliable, effective.The applied method, with the aid of MATHEMATCA or MATLAB, can be easily extended to all kinds of nonlinear evolution equations in mathematical physics.