Exp-function method for Riccati equation and new exact solutions with two arbitrary functions of (2 + 1)-dimensional Konopelchenko–Dubrovsky equations

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Abstract

In this paper, new exact solutions with two arbitrary functions of the (2 + 1)-dimensional Konopelchenko–Dubrovsky equations are obtained by means of the Riccati equation and its generalized solitary wave solutions constructed by the Exp-function method. It is shown that the Exp-function method provides us with a straightforward and important mathematical tool for solving nonlinear evolution equations in mathematical physics.

Introduction

It is well known that nonlinear evolution equations (NLEEs) are often presented to describe the motion of the isolated waves, localized in a small part of space, in many fields like hydrodynamic, plasma physics and nonlinear optic. The investigation of exact solutions of NLEEs plays an important role in the study of these nonlinear physical phenomena and gradually becomes one of the most important and significant tasks. In the past several decades, many effective methods for obtaining exact solutions of NLEEs have been proposed, such as Hirota’s bilinear method [1], Bäcklund transformation [2], Painlevé expansion [3], sine–cosine method [4], homogeneous balance method [5], homotopy perturbation method [6], [7], [8], variational iteration method [9], [10], [11], [12], asymptotic methods [13], non-perturbative methods [14], Adomian decomposition method [15], tanh-function method [16], [17], [18], [19], [20], algebraic method [21], [22], [23], [24], Jacobi elliptic function expansion method [25], [26], [27], F-expansion method [28], [29], [30], [31], [32], [33], [34], [35] and auxiliary equation method [36], [37], [38], [39], [40], [41]. Generally speaking, exact solutions of NLEEs obtained by most of these methods are written as a polynomial in several elementary or special functions which satisfy a first-order ordinary differential equation called sub-equation. It is obvious that the more solutions we find for such sub-equations as sine–Gordon equation, elliptic equation and Riccati equation, the more exact solutions we may obtain for the considered NLEEs.

Recently, He and Wu [42] proposed a direct method called the Exp-function method to obtain exact solutions of NLEEs. The Exp-function method with the help of Matlab or Mathematica is simple and effective, and it has been successfully applied to many kinds of NLEEs [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57], [58], [59], [60] for finding generalized solitary wave solutions and periodic solutions. By setting appropriate values to the arbitrary parameters involved in the generalized solitary wave solutions, we can recover some known solutions, such as those [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57], [58], [59], [60] obtained by the Adomian decomposition method, tanh-function method, algebraic method, Jacobi elliptic function expansion method, F-expansion method and auxiliary equation method. As pointed out in [47], classical Jacobi elliptic function expansion method, tanh-function method and F-expansion method cannot be used to solve NLLEs in which the odd and even-order derivative terms coexist. Fortunately, Ebaid [47] took Burgers equation as an example to illustrate that the Exp-function method is available for such equations. In addition, the Exp-function method for discrete equations [48], [49], [50] is more powerful than the hyperbolic function method [61].

Wu and He [46] applied the Exp-function method to a general Riccati equation and obtained a generalized solitary wave solution, from which all solutions given by Yomba [22] were recovered. In a more recent paper [60], we applied the Exp-function method to the Riccati equation [62]ϕ(ξ)=ddξϕ(ξ)=q+pϕ2(ξ),and obtained three generalized solitary wave solutions. By using the Riccati equation (1) and its generalized solitary wave solutions, we then found some new and more general exact solutions of the (2 + 1)-dimensional Broer–Kaup–Kupershmidt (BKK) equations [60]. It shows that the Exp-function method plays an important role in finding these new and more general exact solutions.

The present paper is motivated by the desire to extend the work [60] to the following (2 + 1)-dimensional Konopelchenko–Dubrovsky (KD) equations: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 solutions were reported in [20], [29], [30], [31], [63].

Section snippets

Exp-function method for Riccati equation

In this section, we recall the work [60] of applying the Exp-function method to the Riccati equation (1).

Introducing a complex variable η defined asη=kξ+ω,where k is a constant to be determined later, ω is an arbitrary constant, Eq. (1) becomeskϕ-q-pϕ2=0,where prime denotes the derivative with respect to η.

According to the Exp-function method [42], we assume that the solution of Eq. (5) can be expressed in the following formϕ(η)=acexp(cη)++a-dexp(-dη)bfexp(fη)++b-gexp(-gη),where c, d, f and g

Exact solutions of the (2 + 1)-dimensional KD equations

Balancing the highest order partial derivative with the nonlinear term [63], we suppose that Eqs. (2), (3) have the following formal solutions:u=a0(y,t)+a-1(y,t)ϕ-1(ξ)+a1(y,t)ϕ(ξ),v=b0(y,t)+b-1(y,t)ϕ-1(ξ)+b1(y,t)ϕ(ξ),where ϕ(ξ) satisfies the Riccati equation (1), ξ=kx+η(y,t),a0(y,t),a-1(y,t),a1(y,t),b0(y,t),b-1(y,t),b1(y,t) and η(y,t) are functions to be determined later, k is a nonzero constant.

Substituting (27), (28) along with Eq. (1) into Eqs. (2), (3), the left-hand sides of Eqs. (2), (3)

Conclusion

Several new exact solutions with two arbitrary functions of the (2 + 1)-dimensional KD equations have been obtained owing to the Riccati equation and its generalized solitary wave solutions constructed by the Exp-function method. The arbitrary functions imply that these solutions have rich spatial structures. It may be important to explain some physical phenomena. The solution procedure used for finding exact solutions of the (2 + 1)-dimensional KD equations is also available for many other NLEEs.

Acknowledgments

I would like to express my sincere thanks to editors and referees for their valuable suggestions and comments. This work was supported by the Natural Science Foundation of Educational Committee of Liaoning Province of China.

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