Zhu Jia-Min

Hyperbolic function method for solving nonlinear differential-different equations

An algorithm is devised to obtained exact travelling wave solutions of differential-different equations by means of hyperbolic function. For illustration, we apply the method to solve the discrete nonlinear (2+1)-dimensional Toda lattice equation and the discretized nonlinear mKdV lattice equation and successfully constructed some explicit and exact travelling wave solutions.

An extended Jacobian elliptic function method for the discrete mKdV lattice

In this paper, the discrete mKdV lattice is solved by using the modified Jacobian elliptic function expansion method. As a consequence, abundant families of Jacobian elliptic function solutions are obtained. When the modulus m→1, these periodic solutions degenerate to the corresponding solitary wave solutions, including bell-type and kink-type excitations.

Fractal Dromion, Fractal Lump, and Multiple Peakon Excitations in a New (2 + 1)-Dimensional Long Dispersive Wave System*

By means of variable separation approach, quite a general excitation of the new (2 + 1)-dimensional long dispersive wave system: is derived. Some types of the usual localized excitations such as dromions, lumps, rings, and oscillating soliton excitations can be easily constructed by selecting the arbitrary functions appropriately. Besides these usual localized structures, some new localized excitations like fractal-dromion, fractal-lump, and multi-peakon excitations of this new system are found by selecting appropriate functions.

General Jacobian elliptic function expansion method and its applications

An extended Jacobian elliptic function expansion method is presented and successfully applied to the nonlinear Schrodinger (NLS) equation and Zakharov equation. We obtain some new solutions besides Fu et als results. The results show that our method is more powerful to construct Jacobian elliptic function and can be applied to other nonlinear physics systems.

New Exact Solutions to （2＋1）-Dimensional Variable Coefficients Broer-Kaup Equations

In this paper, using the variable coefficient generalized projected Ricatti equation expansion method, we present explicit solutions of the (2+1)-dimensional variable coefficients Broer–Kaup (VCBK) equations. These solutions include Weierstrass function solution, solitary wave solutions, soliton-like solutions and trigonometric function solutions. Among these solutions, some are found for the first time. Because of the three or four arbitrary functions, rich localized excitations can be found.

Interaction between two folded solitary waves for a modified Broer–Kaup system

By means of a Painleve–Backlund transformation and a multi-linear separation-of-variable approach, abundant localized coherent excitations of a modified Broer–Kaup system are derived. There appear possible phase shifts for the interactions of the (2+1)-dimensional novel localized structures, which are discussed in this paper.

New Fractal Localized Structures in Boiti-Leon-Pempinelli System

A novel phenomenon that the localized coherent structures of a (2+1)-dimensional physical model possess fractal behaviors is revealed. To clarify the interesting phenomenon, we take the (2+1)-dimensional Boiti–Leon–Pempinelli system as a concrete example. Starting from an extended homogeneous balance approach, a general solution of the system is derived. From which some special localized excitations with fractal behaviors are obtained by introducing some types of lower-dimensional fractal patterns.

Solitary Wave and Periodic Wave Solutions for the Relativistic Toda Lattices

In this work, an adaptation of the tanh /tan-method that is discussed usually in the nonlinear partial differential equations is presented to solve nonlinear polynomial differential-difference equations. As a concrete example, several solitary wave and periodic wave solutions for the chain which is related to the relativistic Toda lattice are derived. Some systems of the differential-difference equations that can be solved using our approach are listed and a discussion is given in conclusion.