Exact Travelling Wave Solutions to a Coupled Nonlinear Evolution Equation

By using an improved hyperbola function method, several types of exact travelling wave solutions to a coupled nonlinear evolution equation are obtained, which include kink-shaped soliton solutions, bell-shaped soliton solutions, envelop solitary wave solutions, and new solitary waves. The method can be applied to other nonlinear evolution equations in mathematical physics.     

扩展的双曲函数法和Zakharov方程组的新精确孤立波解

借助于符号计算软件Maple，利用扩展的双曲函数法求出了Zakharov方程组的精确孤立波解，包括钟状孤立波解、扭结状孤立波解、包络孤立波解、奇性孤立波解和一种新的形式的孤立波解.这种方法也适用于其他非线性波方程.     

Extended Hyperbola Function Method and Its Application to Nonlinear Equations

An extended hyperbola function method is proposed to construct exact solitary wave solutions to nonlinear wave equation based upon a coupled Riccati equation. It is shown that more new solitary wave solutions can be found by this new method, which include kink-shaped soliton solutions, bell-shaped soliton solutions and new solitary wave.The new method can be applied to other nonlinear equations in mathematical physics.     

Some New Solitary Wave Solutions to a Compound KdV-Burgers Equation

In terms of the solutions of an auxiliary ordinary differential equation, a new algebraic method, which contains the terms of first-order derivative of functions f(ξ), is constructed to explore the new solitary wave solutions for nonlinear evolution equations. The method is applied to a compound KdV-Burgers equation, and abundant new solitary wave solutions are obtained. The algorithm is also applicable to a large variety of nonlinear evolution equations.     

Constructing infinite sequence exact solutions of nonlinear evolution equations

To construct the infinite sequence new exact solutions of nonlinear evolution equations and study the first kind of elliptic function, new solutions and the corresponding Bäcklund transformation of the equation are presented. Based on this, the generalized pentavalent KdV equation and the breaking soliton equation are chosen as applicable examples and infinite sequence smooth soliton solutions, infinite sequence peak solitary wave solutions and infinite sequence compact soliton solutions are obtained with the help of symbolic computation system Mathematica. The method is of significance to search for infinite sequence new exact solutions to other nonlinear evolution equations.     

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.     

广义五阶KdV方程的孤波解与孤子解

利用求解非线性代数方程组的吴文俊特征列方法,借助计算机代数系统获得了一类较广泛的五阶非线性演化方程的孤波解和孤子解,修正和完善了已知的结论 关键词： 五阶KdV方程 孤波 孤子解     

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     

Extended F-Expansion Method and Periodic Wave Solutionsfor Klein-Gordon-Schrödinger Equations

We present an extended F-expansion method for finding periodic wave solutions of nonlinear evolution equations in mathematical physics. By using extended F-expansion method, many periodic wave solutions expressed by various Jacobi elliptic functions for the Klein-Gordon-Schrödinger equations are obtained. In the limit cases, the solitary wave solutions and trigonometric function solutions for the equations are also obtained.     

密度矩阵重正化群的异构并行优化

密度矩阵重正化群方法（DMRG）在求解一维强关联格点模型的基态时可以获得较高的精度,在应用于二维或准二维问题时,要达到类似的精度通常需要较大的计算量与存储空间.本文提出一种新的DMRG异构并行策略,可以同时发挥计算机中央处理器（CPU）和图形处理器（GPU）的计算性能.针对最耗时的哈密顿量对角化部分,实现了数据的分布式存储,并且给出了CPU和GPU之间的负载平衡策略.以费米Hubbard模型为例,测试了异构并行程序在不同DMRG保留状态数下的运行表现,并给出了相应的性能基准.应用于4腿梯子时,观测到了高温超导中常见的电荷密度条纹,此时保留状态数达到10⁴,使用的GPU显存小于12 GB.     

Applications of F-expansion to Periodic Wave Solutions for Variant Boussinesq Equations

We present an F-expansion method for finding periodic wave solutions of nonlinear evolution equations in mathematical physics, which can be thought of as a concentration of extended Jacobi elliptic function expansion method proposed recently. By using the F-expansion, without calculating Jacobi elliptic functions, we obtain simultaneously many periodic wave solutions expressed by various Jacobi elliptic functions for the variant Boussinesq equations. When the modulus m approaches 1 and 0, the hyperbolic function solutions (including the solitary wave solutions) and trigonometric solutions are also given respectively.     

Applications of F-expansion to Periodic Wave Solutions for Variant Boussinesq Equations

We present an F-expansion method for finding periodic wave solutions of nonlinear evolution equations in mathematical physics, which can be thought of as a concentration of extended Jacobi elliptic function expansion method proposed recently. By using the F-expansion, without calculating Jacobi elliptic functions, we obtain simultaneously many periodic wave solutions expressed by various Jacobi elliptic functions for the variant Boussinesq equations. When the modulus m approaches 1 and O, the hyperbolic function solutions （including the solitary wave solutions） and trigonometric solutions are also given respectively.     

Abundant different types of exact soliton solution to the (4+1)-dimensional Fokas and (2+1)-dimensional breaking soliton equations

The prime objective of this paper is to explore the new exact soliton solutions to the higher-dimensional nonlinear Fokas equation and(2+1)-dimensional breaking soliton equations via a generalized exponential rational function(GERF) method. Many different kinds of exact soliton solution are obtained, all of which are completely novel and have never been reported in the literature before. The dynamical behaviors of some obtained exact soliton solutions are also demonstrated by a choice of appropriate values of the free constants that aid in understanding the nonlinear complex phenomena of such equations. These exact soliton solutions are observed in the shapes of different dynamical structures of localized solitary wave solutions, singular-form solitons, single solitons,double solitons, triple solitons, bell-shaped solitons, combo singular solitons, breather-type solitons,elastic interactions between triple solitons and kink waves, and elastic interactions between diverse solitons and kink waves. Because of the reduction in symbolic computation work and the additional constructed closed-form solutions, it is observed that the suggested technique is effective, robust, and straightforward. Moreover, several other types of higher-dimensional nonlinear evolution equation can be solved using the powerful GERF technique.     

Applications of F-expansion method to the coupled KdV system

An extended F-expansion method for finding periodic wave solutions of nonlinear evolution equations in mathematical physics is presented, which can be thought of as a concentration of extended Jacobi elliptic function expansion method proposed more recently. By using the homogeneous balance principle and the extended F-expansion, more periodic wave solutions expressed by Jacobi elliptic functions for the coupled KdV equations are derived. In the limit cases, the solitary wave solutions and the other type of travelling wave solutions for the system are also obtained.     

考虑量子效应的Zakharov方程组的孤波解

借助Maple程序，利用扩展的双曲函数法和双函数法求解了考虑量子效应的Zakharov方程组，得到了多种孤波解，其中包括亮孤波解、W型孤波解、M型孤波解和奇性孤波解. 关键词： 量子效应 Zakharov方程组 扩展的双曲函数法 孤波解     

修正cKdV方程组的孤立波结构及其稳定性

利用函数展开法求解修正耦合KdV(Coupled KdV,cKdV)方程组,得到几类孤立波解,包括扭结型-钟型、双扭结型、双钟型以及双扭结-双钟型结构的单孤子解.在不同的极限情况下,这些解分别退化为修正cKdV方程的扭结状或钟状孤波解.对孤立波的稳定性进行了数值研究,结果表明:修正cKdV方程既存在稳定的孤立波解,也存在不稳定的孤立波解.     

A method for constructing exact solutions and application to Benjamin Ono equation

By using an improved projective Riccati equation method, this paper obtains several types of exact travelling wave solutions to the Benjamin Ono equation which include multiple soliton solutions, periodic soliton solutions and Weierstrass function solutions. Some of them are found for the first time. The method can be applied to other nonlinear evolution equations in mathematical physics.     

Applications of the first integral method to nonlinear evolution equations

<正>In this paper,we establish travelling wave solutions for some nonlinear evolution equations.The first integral method is used to construct the travelling wave solutions of the modified Benjamin-Bona-Mahony and the coupled Klein-Gordon equations.The obtained results include periodic and solitary wave solutions.The first integral method presents a wider applicability to handling nonlinear wave equations.     

双曲函数型辅助方程构造具5次强非线性项的波方程的新精确孤波解

在辅助方程法的基础上利用两种函数变换和一种双曲函数型辅助方程，通过符号计算系统Mathematica构造了在力学当中一个重要的模型，有5次强非线性项的波方程的新三角函数型和双曲函数型精确孤波解.这种方法寻找其他具5次强非线性项的非线性发展方程的新精确解方面具有普遍意义. 关键词： 双曲函数型辅助方程 函数变换 具5次强非线性项的波方程 精确孤波解     

New Modified Jacobi Elliptic Function Expansion Method and Its Application to (3+1)-Dimensional KP Equation

With the aid of computerized symbolic computation, the new modified Jacobi elliptic function expansion method for constructing exact periodic solutions of nonlinear mathematical physics equation is presented by a new general ansatz. The proposed method is more powerful than most of the existing methods. By use of the method, we not only can successfully recover the previously known formal solutions but also can construct new and more general formal solutions for some nonlinear evolution equations. We choose the (3+1)-dimensional Kadomtsev-Petviashvili equation to illustrate our method. As a result, twenty families of periodic solutions are obtained. Of course, more solitary wave solutions, shock wave solutions or triangular function formal solutions can be obtained at their limit condition.