Engui Fan

Extended tanh-function method and its applications to nonlinear equations

Abstract An extended tanh-function method is proposed for constructing multiple travelling wave solutions of nonlinear partial differential equations (PDEs) in a unified way. The key idea of this method is to take full advantages of a Riccati equation involving a parameter and use its solutions to replace the tanh function in the tanh-function method. It is quite interesting that the sign of the parameter can be used to exactly judge the numbers and types of these travelling wave solutions. In addition, by introducing appropriate transformations, it is shown that the extended tanh-function method still is applicable to nonlinear PDEs whose balancing numbers may be any nonzero real numbers. Some illustrative equations are investigated by this means and new travelling wave solutions are found.

Uniformly constructing a series of explicit exact solutions to nonlinear equations in mathematical physics

Abstract In this paper, we devise a new unified algebraic method to construct a series of explicit exact solutions for general nonlinear equations. Compared with most existing methods such as tanh method, Jacobi elliptic function method and homogeneous balance method, the proposed method not only gives new and more general solutions, but also provides a guideline to classify the various types of the solutions according to the values of some parameters. The solutions obtained in this paper include (a) polynomial solutions, (b) exponential solutions, (c) rational solutions, (d) triangular periodic wave solutions, (e) hyperbolic, and soliton solutions, (f) Jacobi, and Weierstrass doubly periodic wave solutions. The efficiency of the method can be demonstrated on a large variety of nonlinear equations such as those considered in this paper, combined KdV–MKdV, Camassa–Holm, Kaup–Kupershmidt, Jaulent–Miodek, (2+1)-dimensional dispersive long wave, new (2+1)-dimensional generalized Hirota, (2+1)-dimensional breaking soliton and double sine-Gordon equations. In addition, the links among our proposed method, the tanh method, the extended method and the Jacobi function expansion method are also clarified generally.

Two new applications of the homogeneous balance method

Abstract The homogeneous balance method is extended to search for Backlund transformation and similarity reduction of nonlinear partial differential equations. It is shown that there exist close connections among the homogeneous balance method, WTC method and CK direct reduction method. The variant Boussinesq equations are discussed as an illustrative example. Their Backlund transformation, linearization transformation and three types of similarity reductions are obtained. In the meantime, some new travelling wave solutions also are found.

Soliton solutions for a generalized Hirota–Satsuma coupled KdV equation and a coupled MKdV equation

Abstract We make use of an extended tanh-function method and symbolic computation to obtain respectively four kinds of soliton solutions for a new generalized Hirota–Satsuma coupled KdV equation and a new coupled MKdV equation, which were introduced recently by Wu et al. (Phys. Lett. A 255 (1999) 259).

Integrable evolution systems based on Gerdjikov–Ivanov equations, bi-Hamiltonian structure, finite-dimensional integrable systems and N-fold Darboux transformation

A spectral problem and the associated Gerdjikov–Ivanov (GI) hierarchy of nonlinear evolution equations is presented. As a reduction, the well-known GI equation of derivative nonlinear Schrodinger equations is obtained. It is shown that the GI hierarchy is integrable in a Liouville sense and possesses bi-Hamiltonian structure. Moreover, the spectral problem can be nonlinearized as a finite dimensional completely integrable system under the Bargmann constraint between the potentials and the eigenfunctions. In particular, an explicit N-fold Darboux transformation for the GI equation is constructed with the help of a gauge transformation of spectral problems and a reduction technique. Some explicit solitonlike solutions of the GI equation are given by applying its Darboux transformation.

Auto-Bäcklund transformation and similarity reductions for general variable coefficient KdV equations

Abstract Based on the idea of the homogeneous balance (HB) method, we study the Backlund transformation and similarity reductions of general variable coefficient KdV equation. It is found that the corresponding results are coincide with those by Weiss–Tabor–Carnevale (WTC) method and Clarkson–Kruskal (CK) method, respectively. The close connections among the HB, WTC and CK methods are exposured in theory. In the meantime, a Lax pair, a symmetry, two conservation laws and a analytic solution for the general variable coefficient KdV equation are given.

Generalized tanh Method Extended to Special Types of Nonlinear Equations

By some ‘pre-possessing’ techniques we extend the generalized tanh method to special types of nonlinear equations for constructing their multiple travelling wave solutions. The efficiency of the method can be demonstrated for a large variety of special equations such as those considered in this paper, double sine-Gordon equation, (2+1)-dimensional sine-Gordon equation, Dodd-Bullough- Mikhailov equation, coupled Schrödinger-KdV equation and (2+1)-dimensional coupled Davey- Stewartson equation. - Pacs: 03.40.Kf; 02.30.Jr.

Applications of extended tanh method to 'special' types of nonlinear equations

In this paper, we explore more applications of extended tanh method to some special nonlinear equations. Such equations are not of differential polynomial form so that cannot be directly dealt with by tanh method or extended tanh method. As examples, we apply a recently proposed extended tanh method to build some new explicit solutions for (2+1)-dimensional sine-Gordon equation, Dodd-Bullough-Mikhailov equation and coupled Schrodinger-KdV equation.

Travelling wave solutions in terms of special functions for nonlinear coupled evolution systems

We present a simple and effective transformation that decompose the solution of a coupled system into solving a set of algebraic equations and a first order ordinary differential equation of a certain unknown function. This method not only gives us a clear relation between unknown functions for a nonlinear coupled system, but also easily provides us with a series of new and more general travelling wave solutions in terms of special functions such as hyperbolic, rational, triangular, Weierstrass and Jacobi elliptic double periodic functions.

Double periodic solutions with Jacobi elliptic functions for two generalized Hirota–Satsuma coupled KdV systems

Abstract We make use of Jacobi elliptic functions to construct new double periodic wave solutions for two generalized Hirota–Satsuma coupled KdV systems. It is shown that these solutions exactly degenerate to the well-known soliton solutions at a limit condition.