A generalized F-expansion method and new exact solutions of Konopelchenko–Dubrovsky equations

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Abstract

In this paper, a generalized F-expansion method is proposed to seek more general exact solutions of nonlinear partial differential equations. With the aid of symbolic computation, we choose the (2 + 1)-dimensional Konopelchenko–Dubrovsky equations to illustrate the validity and advantages of the method. As a result, many new and more general exact solutions are obtained including single and combined Jacobi elliptic function solutions, soliton-like solutions, trigonometric function solutions. The method can be applied to other nonlinear partial differential equations in mathematical physics.

Introduction

In recent years, nonlinear partial differential equations (NLPDEs) are widely used to describe many important phenomena and dynamic processes in physics, mechanics, chemistry, biology, etc. As mathematical models of the phenomena, the investigation of exact solutions of NLPDEs will help one to understand these phenomena better. Many powerful methods for obtaining exact solutions of NLPDEs have been presented, such as inverse scattering method [1], Hirota’s bilinear method [2], Bäcklund transformation [3], Painlevé expansion [4], sine–cosine method [5], homotopy perturbation method [6], [7], [8], Adomian Pade approximation [9], homogenous balance method [10], variational method [11], [12], [13], [14], algebraic method [15], tanh function method [16], [17], [18], [19], [20], [21], [22] and so on. Recently F-expansion method [23], [24], [25], [26] was proposed to construct periodic wave solutions of NLPDEs, which can be thought of as an over-all generalization of Jacobi elliptic function expansion method [27], [28], [29], [30]. F-expansion method was later extended in different manners [31], [32], [33], [34], [35].

In this paper, by using a new and more general ansatz solution we propose a generalized F-expansion method to construct more general exact solutions of NLPDEs. In order to illustrate the validity and advantages of the method, the (2 + 1)-dimensional Konopelchenko–Dubrovsky (KD) equations are considered. As a result, many new and more general non-travelling wave and coefficient function solutions are successfully obtained including single and combined non-degenerate Jacobi elliptic function solutions, soliton-like solutions, trigonometric function solutions.

The rest of this paper is organized as follows: in Section 2, we give the description of the generalized F-expansion method; in Section 3, we apply this method to the (2 + 1)-dimensional KD equations; in Section 4, some conclusions are given.

Section snippets

Description of the generalized F-expansion method

For a given NLPDE with independent variables x = (t, x1, x2,  , xm) and dependent variable u:F(u,ut,ux1,ux2,,uxm,ux1t,ux2t,uxmt,utt,ux1x1,ux2x2,,uxmxm,)=0,we seek its solutions in the new and more general form:u=a0+i=1n{aiF-i(ω)+biFi(ω)+ciFi-1(ω)F(ω)+diF-i(ω)F(ω)}.where a0 = a0(x), ai = ai(x), bi = bi(x), ci = ci(x), di = di(x) (i = 1, 2,  , n) and ω = ω(x) are all functions to be determined, F(ω) and F′(ω) in (2) satisfyF2(ω)=PF4(ω)+QF2(ω)+R,and hence holds for F(ω) and F′(ω)F(ω)=2PF3(ω)+QF(ω),F(3)(ω)=(6PF2(

New exact solutions of the KD equations

Let us consider the (2 + 1)-dimensional KD equation [34], [36]:ut-uxxx-6buux+32a2u2ux-3vy+3auxv=0,uy=vx,where a and b are real constants. Jacobi elliptic function solutions, soliton-like solutions, trigonometric function solutions and rational function solutions can be found in Refs. [34], [36]. In this section, we will apply our method to obtain new and more general exact solutions of Eqs. (5), (6).

According to Step 1, we get n = 1 for u and v. In order to search for explicit solutions of Eqs. (5)

Conclusion

In this paper, we have proposed a generalized F-expansion method to seek more general exact solutions of NLPDEs. With the aid of Mathematica, the method provides a powerful mathematical tool to obtain more general exact solutions of a great many NLPDEs in mathematical physics, such as the (3 + 1)-dimensional Kadomtsev–Petviashvili (KP) equation, the (2 + 1)-dimensional Broer–Kaup–Kupershmidt (BKK) equations, breaking soliton (BS) equations, Nizhnik–Novikov–Vesselov (NNV) equations, dispersive long

Acknowledgements

The authors would like to express their thanks to referee for the valuable advice. This work was supported by the Natural Science Foundation of Educational Committee of Liaoning Province of China.

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