Li Zhi-Bin

An Automated Algebraic Method for Finding a Series of Exact Travelling Wave Solutions of Nonlinear Evolution Equations

Based on a type of elliptic equation, a new algebraic method to construct a series of exact solutions for nonlinear evolution equations is proposed, meanwhile, its complete implementation TRWS in Maple is presented. The TRWS can output a series of travelling wave solutions entirely automatically, which include polynomial solutions, exponential function solutions, triangular function solutions, hyperbolic function solutions, rational function solutions, Jacobi elliptic function solutions, and Weierstrass elliptic function solutions. The effectiveness of the package is illustrated by applying it to a variety of equations. Not only are previously known solutions recovered but also new solutions and more general form of solutions are obtained.

An automated Jacobi elliptic function method for finding periodic wave solutions to nonlinear evolution equations

We describe the Jacobi elliptic function method for finding exact periodic wave solutions to nonlinear evolution equations. We present a Maple packaged automated Jacobi elliptic function method, which can entirely automatically output the exact periodic wave solutions. The effectiveness of the automated Jacobi elliptic function method is demonstrated using as examples the application to a variety of equations with physical interest. Not only are the previously known solutions recovered but in some cases new solutions and more general forms of solutions are obtained.

Darboux Transformation and Multi-Solitons for Complex mKdV Equation

An explicit N-fold Darboux transformation with multi-parameters for coupled mKdV equation is constructed with the help of a gauge transformation of the Ablowitz-Kaup-Newell-Segur (AKNS) system spectral problem. By using the Darboux transformation and the reduction technique, some multi-soliton solutions for the complex mKdV equation are obtained.

Applications of Jacobi Elliptic Function Expansion Method for Nonlinear Differential-Difference Equations

The Jacobi elliptic function expansion method is extended to derive the explicit periodic wave solutions for nonlinear differential-difference equations. Three well-known examples are chosen to illustrate the application of the Jacobi elliptic function expansion method. As a result, three types of periodic wave solutions including Jacobi elliptic sine function, Jacobi elliptic cosine function and the third elliptic function solutions are obtained. It is shown that the shock wave solutions and solitary wave solutions can be obtained at their limit condition.

Conservation laws and new exact solutions for the generalized seventh order KdV equation

Abundant polynomial type conservation laws are constructed for a seventh order nonlinear evolution equation with six arbitrary parameters. From the parameters constraints that leading to the existence of conserved densities, a new unnamed seventh order KdV type equation that may be integrable is reported. By introducing nonlinear transformation, the new soliton solution as well as the solitary wave solution to the new unnamed seventh order KdV type equation are obtained.

New Multi-soliton Solutions for the (2+1)-Dimensional Kadomtsev–Petviashvili Equation

In this paper, an explicit N-fold Darboux transformation with multi-parameters for both a (1+1)-dimensional Broer–Kaup (BK) equation and a (1+1)-dimensional high-order Broer–Kaup equation is constructed with the help of a gauge transformation of their spectral problems. By using the Darboux transformation and new basic solutions of the spectral problems, 2N-soliton solutions of the BK equation, the high-order BK equation, and the Kadomtsev–Petviashvili (KP) equation are obtained.

A Unified Explicit Construction of 2N-Soliton Solutions for Evolution Equations Determined by 2 × 2 AKNS System*

An explicit N-fold Darboux transformation for evolution equations determined by general 2 × 2 AKNS system is constructed. By using the Darboux transformation, the solutions of the evolution equations are reduced to solving a linear algebraic system, from which a unified and explicit formulation of 2N-soliton solutions for the evolution equation are given. Furthermore, a reduction technique for MKdV equation is presented, and an N-fold Darboux transformation of MKdV hierarchy is constructed through the reduction technique. A Maple package which can entirely automatically output the exact N-soliton solutions of the MKdV equation is developed.

A Maple Package for the Painlevé Test of Nonlinear Partial Differential Equations

A Maple package, named PLtest, is presented to study whether or not nonlinear partial differential equations (PDEs) pass the Painleve test. This package is based on the so-called WTC-Kruskal algorithm, which combines the standard WTC algorithm and the Kruskal simplification algorithm. Therefore, we not only study whether the given PDEs pass the test or not, but also obtain its truncated expansion form related to some integrability properties. Several well-known nonlinear models with physical interests illustrate the effectiveness of this package.

Adomian decomposition method and Padé approximants for solving the Blaszak–Marciniak lattice

The Adomian decomposition method (ADM) and Pade approximants are combined to solve the well-known Blaszak–Marciniak lattice, which has rich mathematical structures and many important applications in physics and mathematics. In some cases, the truncated series solution of ADM is adequate only in a small region when the exact solution is not reached. To overcome the drawback, the Pade approximants, which have the advantage in turning the polynomials approximation into a rational function, are applied to the series solution to improve the accuracy and enlarge the convergence domain. By using the ADM-Pade technique, the soliton solutions of the Blaszak–Marciniak lattice are constructed with better accuracy and better convergence than by using the ADM alone. Numerical and figurative illustrations show that it is a promising tool for solving nonlinear problems.

Explicit solutions to the coupled KdV equations with variable coefficients

By means of sn-function expansion method and cn-function expansion method, several kinds of explicit solutions to the coupled KdV equations with variable coefficients are obtained, which include three sets of periodic wave-like solutions. These solutions degenerate to solitary wave-like solutions at a certain limit. Some new solutions are presented.