用推广的Jacobian椭圆函数方法解离散的mKdV格子

本文用推广的Jacobian椭圆函数方法求解了离散的mKdV格子，得到了Jacobian椭圆函数双周期解，当模取1时，可得到钟型孤波解和冲击型孤波解。     

Symbolic Computation of Extended Jacobian Elliptic Function Algorithm for Nonlinear Differential-Different Equations

The Jacobian elliptic function expansion method for nonlinear differential-different equations and its algorithm are presented by using some relations among ten Jacobian elliptic functions and successfully construct more new exact doubly-periodic solutions of the integrable discrete nonlinear Schr ödinger equation. When the modulous m→1 or 0, doubly-periodic solutions degenerate to solitonic solutions including bright soliton, dark soliton, new solitons as well as trigonometric function solutions.     

New Jacobian Elliptic Function Solutions of Modified KdV Equation: Ⅱ

Recently, we obtained thirteen families of Jacobian elliptic function solutions of mKdV equation by usingour extended Jacobian elliptic function expansion method. In this note, the mKdV equation is investigated and anotherthree families of new doubly periodic solutions (Jacobian elliptic function solutions) are fbund again by using a newtransformation, which and our extended Jacobian elliptic function expansion method form a new method still called theextended Jacobian elliptic function expansion method. The new method can be more powertul to be applied to othernonlinear differential equations.     

试探方程法及其在非线性发展方程中的应用

提出了一种比较系统的求解非线性发展方程精确解的新方法, 即试探方程法. 以一个带5阶 导数项的非线性发展方程为例, 利用试探方程法化成初等积分形式，再利用三阶多项式的完 全判别系统求解，由此求得的精确解包括有理函数型解, 孤波解, 三角函数型周期解, 多项 式型Jacobi椭圆函数周期解和分式型Jacobi椭圆函数周期解 关键词： 试探方程法 非线性发展方程 孤波解 Jacobi椭圆函数 周期解     

New Jacobian Elliptic Function Solutions of Modified KdV Equation: II

Recently, we obtained thirteen families of Jacobian elliptic function solutions of mKdV equation by using our extended Jacobian elliptic function expansion method. In this note, the mKdV equation is investigated and another three families of new doubly periodic solutions (Jacobian elliptic function solutions) are found again by using a new transformation, which and our extended Jacobian elliptic function expansion method form a new method still called the extended Jacobian elliptic function expansion method. The new method can be more powerful to be applied to other nonlinear differential equations.     

A general mapping approach and new travelling wave solutions to the general variable coefficient KdV equation

A general mapping deformation method is applied to a generalized variable coefficient KdV equation. Many new types of exact solutions, including solitary wave solutions, periodic wave solutions, Jacobian and Weierstrass doubly periodic wave solutions and other exact excitations are obtained by the use of a simple algebraic transformation relation between the generalized variable coefficient KdV equation and a generalized cubic nonlinear Klein－Gordon equation.     

A General Mapping Approach and New Travelling Wave Solutions to (2+1)-Dimensional Boussinesq Equation

A general mapping deformation method is presented and applied to a (2+1)-dimensional Boussinesq system. Many new types of explicit and exact travelling wave solutions, which contain solitary wave solutions, periodic wave solutions, Jacobian and Weierstrass doubly periodic wave solutions, and other exact excitations like polynomial solutions, exponential solutions, and rational solutions, etc., are obtained by a simple algebraic transformation relation between the (2+1)-dimensional Boussinesq equation and a generalized cubic nonlinear Klein-Gordon equation.     

Rational Form Solitary Wave Solutions and Doubly Periodic Wave Solutions to(1+1)-Dimensional Dispersive Long Wave Equation

In this work we devise an algebraic method to uniformly construct rational form solitary wave solutions and Jacobi and Weierstrass doubly periodic wave solutions of physical interest for nonlinear evolution equations. With the aid of symbolic computation, we apply the proposed method to solving the (1+1)-dimensional dispersive long wave equation and explicitly construct a series of exact solutions which include the rational form solitary wave solutions and elliptic doubly periodic wave solutions as special cases.     

An Algebraic Method for Constructing Exact Solutions to Difference-Differential Equations

In this paper, we present a method to solve difference differential equation(s). As an example, we apply this method to discrete KdV equation and Ablowitz-Ladik lattice equation. As a result, many exact solutions are obtained with the help of Maple including soliton solutions presented by hyperbolic functions sinh and cosh, periodic solutions presented by sin and cos and rational solutions. This method can also be used to other nonlinear difference-differential equation(s).     

Lie Group of Transformations for a KdV–Boussinesq Equation

We consider a combined Korteweg–deVries and Boussinesq equation governing long surface waves in shallow water. Considering traveling wave solutions, the basic equations will be reduced to a second order ordinary differential equation. Using the Lie group of transformations we reduce it to a first order ordinary differential equation and employ a direct method to derive its periodic solutions in terms of Jacobian elliptic functions and their corresponding solitary wave and explode decay mode solutions.     

Doubly Periodic Wave Solutions and Soliton Solutions of Ablowitz–Ladik Lattice System

The general Jacobi elliptic function expansion method is developed and extended to construct doubly periodic wave solutions for discrete nonlinear equations. Applying this method, many exact elliptic function doubly periodic wave solutions are obtained for Ablowitz–Ladik lattice system. When the modulus m→1 or m→0, these solutions degenerate into hyperbolic function solutions and trigonometric function solutions respectively. In long wave limit, solitonic solutions including bright soliton and dark soliton solutions are also obtained.     

Periodic Waves of a Discrete Higher Order Nonlinear SchrSdinger Equation

The Hirota equation is a higher order extension of the nonlinear Schr6dinger equation by incorporating third order dispersion and one form of self steepening effect, New periodic waves for the discrete Hirota equation are given in terms of elliptic functions. The continuum limit converges to the analogous result for the continuous Hirota equation, while the long wave limit of these discrete periodic patterns reproduces the known resulr of the integrable Ablowitz-Ladik system.     

A Study for Obtaining New and More General Solutions of Special-Type Nonlinear Equation

The generalized algebraic method with symbolic computation is extended to some special-type nonlinear equations for constructing a series of new and more general travelling wave solutions in terms of special functions. Such equations cannot be directly dealt with by the method and require some kinds of pre-processing techniques. It is shown that soliton solutions and triangular periodic solutions can be established as the limits of the Jacobi doubly periodic wave solutions.     

New exact solutions to some difference differential equations

In this paper, we use our method to solve the extended Lotka--Volterra equation and discrete KdV equation. With the help of Maple, we obtain a number of exact solutions to the two equations including soliton solutions presented by hyperbolic functions of \sinh and \cosh, periodic solutions presented by trigonometric functions of \sin and \cos, and rational solutions. This method can be used to solve some other nonlinear difference--differential equations.     

A Series of Exact Solutions for a New (2+1)-Dimensional Calogero KdV Equation

An algebraic method is proposed to solve a new (2 1)-dimensional Calogero KdV equation and explicitly construct a series of exact solutions including rational solutions, triangular solutions, exponential solution, line soliton solutions, and doubly periodic wave solutions.     

A Series of Exact Solutions for a New （2＋1）-Dimensional Calogero KdV Equation

An algebraic method is proposed to solve a new （2＋1）-dimensional Calogero KdV equation and explicitly construct a series of exact solutions including rational solutions, triangular solutions, exponential solution, line soliton solutions, and doubly periodic wave solutions.     

A New Rational Algebraic Approach to Find Exact Analytical Solutions to a （2＋1）-Dimensional System

In this paper, we present a new rational algebraic approach to uniformly construct a series of exact analytical solutions for nonlinear partial differential equations. Compared with most existing tanh methods and other sophisticated methods, the proposed method not only recovers some known solutions, but also finds some new and general solutions. The solutions obtained in this paper include rational form triangular periodic wave solutions, solitary wave solutions, and elliptic doubly periodic wave solutions. The efficiency of the method can be demonstrated on （2＋1）-dimensional dispersive long-wave equation.     

Complex dynamics analysis of impulsively coupled Duffing oscillators with ring structure

Impulsively coupled systems are high-dimensional non-smooth systems that can exhibit rich and complex dynamics.This paper studies the complex dynamics of a non-smooth system which is unidirectionally impulsively coupled by three Duffing oscillators in a ring structure.By constructing a proper Poincare map of the non-smooth system,an analytical expression of the Jacobian matrix of Poincare map is given.Two-parameter Hopf bifurcation sets are obtained by combining the shooting method and the Runge-Kutta method.When the period is fixed and the coupling strength changes,the system undergoes stable,periodic,quasi-periodic,and hyper-chaotic solutions,etc.Floquet theory is used to study the stability of the periodic solutions of the system and their bifurcations.     

Doubly periodic waves of a discrete nonlinear Schrödinger system with saturable nonlinearity

Abstract

A system of two discrete nonlinear Schrödinger equations of the Ablowitz-Ladik type with a saturable nonlinearity is shown to admit a doubly periodic wave, whose long wave limit is also derived. As a by-product, several new solutions of the elliptic type are provided for NLS-type discrete and continuous systems.