Hongqing Zhang

A note on the homogeneous balance method

Abstract Based on the idea of the homogeneous balance method, a simple and efficient method is proposed for obtaining exact solutions of nonlinear partial differential equations. Some equations are investigated by this means and new solitary wave solutions or singular traveling wave solutions are found.

New explicit solitary wave solutions and periodic wave solutions for Whitham–Broer–Kaup equation in shallow water

Abstract Based upon the well-known Riccati equation, a new generalized transformation is presented and applied to solve Whitham–Broer–Kaup (WBK) equation in shallow water. As a result, many explicit exact solutions, which contain new solitary wave solutions, periodic wave solutions and the combined formal solitary wave solutions and periodic wave solutions, are obtained. And variant Boussinesq equation and the system of approximate equation for long water waves, as the special cases of WBK equation, can also obtain the corresponding solitary wave solutions and periodic wave solutions. In addition, with the aid of Mathematica and Wu elimination method to solve a large system of algebraic equations, the course of solving equations can be carried out in computer.

New explicit and exact travelling wave solutions for a system of variant Boussinesq equations in mathematical physics

Abstract With the aid of Mathematica, we obtain several types of explicit and exact travelling wave solutions to a system of variant Boussinesq equations by using an improved sine-cosine method and the Wu elimination method. These solutions contain Wangs results and other types of solitary wave solutions and new solutions. The method can also be applied to solve more systems of nonlinear partial differential equations, such as the coupled KdV equations.

Symbolic computation and new families of exact soliton-like solutions to the integrable Broer-Kaup (BK) equations in (2+1)-dimensional spaces

In this paper, new families of soliton-like solutions are obtained for (2+1)-dimensional integrable Broer-Kaup equations by using the symbolic computation method developed by Gao and Tian. Sample solutions obtained from these methods are presented. Solitary wave solutions are merely a special case in one family. The method can also be extended to other types of nonlinear evolution equations in mathematical physics.

New exact solutions to a system of coupled KdV equations

Abstract Using an improved homogeneous balance method, several kinds of exact solutions which include Wangs results are obtained for a system of coupled KdV equations.

Explicit exact solutions for compound KdV-type and compound KdV–Burgers-type equations with nonlinear terms of any order

Abstract In this paper, we improved a method presented previously (Phys. Lett. A 285 (2001) 355) by means of a proper transformation. Applying the improved method, we consider the generalized compound KdV-type and compound KdV–Burgers-type equations with nonlinear terms of any order. As a result, many explicit exact solutions, which contain new solitary wave solutions, periodic wave solutions and the combined formal solitary wave solutions, are obtained.

Exact travelling wave solutions for a generalized Zakharov-Kuznetsov equation

In this paper, by means of a proper transformation and symbolic computation, we study the exact travelling wave solutions for a generalized Zakharov-Kuznetsov (GZK) equation by using the extended-tanh method and direct assumption method. As a result, rich exact travelling wave solutions, which contain new kink-shaped solitons, bell-shaped solitons, periodic solutions, combined formal solitons, rational solutions and singular solitons for GZK equation, are obtained.

New multiple soliton solutions to the general Burgers-Fisher equation and the Kuramoto-Sivashinsky equation

Abstract A generalized tanh function method is used for constructing exact travelling wave solutions of nonlinear partial differential equations in a unified way. The main idea of this method is to take full advantage of the Riccati equation which has more new solutions. More new multiple soliton solutions are obtained for the general Burgers–Fisher equation and the Kuramoto–Sivashinsky equation.

On the integrability of a generalized variable-coefficient Kadomtsev–Petviashvili equation

By considering the inhomogeneities of media, a generalized variable-coefficient Kadomtsev?Petviashvili (vc-KP) equation is investigated, which can be used to describe many nonlinear phenomena in fluid dynamics and plasma physics. In this paper, we systematically investigate the complete integrability of the generalized vc-KP equation under an integrable constraint condition. With the aid of generalized Bell?s polynomials, its bilinear formalism, bilinear B?cklund transformations, Lax pairs and Darboux covariant Lax pairs are succinctly constructed, which can be reduced to the ones of several integrable equations such as KdV, cylindrical KdV, KP, cylindrical KP, generalized cylindrical KP, non-isospectral KP equations, etc. Moreover, the infinite conservation laws of the equation are found by using its Lax equations. All conserved densities and fluxes are given with explicit recursion formulas. Furthermore, an extra auxiliary variable is introduced to obtain the bilinear formalism, based on which, the soliton solutions and Riemann theta function periodic wave solutions are presented. The influence of inhomogeneity coefficients on solitonic structures and interaction properties are discussed for physical interest and possible applications by some graphic analysis. Finally, a limiting procedure is presented to analyze in detail the asymptotic behavior of the periodic waves and the relations between the periodic wave solutions and soliton solutions.

Explicit and exact traveling wave solutions of Whitham–Broer–Kaup shallow water equations

Abstract In this Letter, four pairs solutions of Whitham–Broer–Kaup (WBK) equations, which contain blow-up solutions and periodic solutions, are obtained by using of hyperbolic function method, Mathematica and Wu elimination method. The method can also be applied to solve more nonlinear partial differential equation or equations.