BIFURCATIONS OF TRAVELLING WAVE SOLUTIONS TO A COUPLED NONLINEAR WAVE SYSTEM

Employ theory of bifurcations of dynamical systems to a system of coupled nonlin-ear equations, the existence of solitary wave solutions, kink wave solutions, anti-kink wave solutions and periodic wave solutions is obtained. Under different parametric conditions, various suffcient conditions to guarantee the existence of the above so-lutions are given. Some exact explicit parametric representations of travelling wave solutions are derived.     

The bifurcation and exact travelling wave solutions for the modified Benjamin-Bona-Mahoney (mBBM) equation

By using methods from dynamical systems theory, this paper researches the bifurcation and exact travelling wave solutions for the modified Benjamin-Bona-Mahoney (mBBM) equation. Implicit exact parametric representations of all travelling wave solutions as well as some explicit analytic solutions are given. Specially, breaking wave solutions are obtained, which KdV equation does not include.     

齐次平衡法和浅水波中WBK方程的精确分析解

Based on the homogenous balance method and with the help of mathematica, the Backlund transformation and the transfer heat equation are derived. Analyzing the heat-transfer equation, the multiple soliton solutions and other exact analytical solution for Whitham-Broer-Kaup equations(WBK) are derived. These solutions contain Fan‘s, Xie‘s and Yan‘s results and other new types of analytical solutions, such as rational function solutions and periodic solutions. The method can also be applied to solve more nonlinear differential equations.     

矩阵方程AX=A+X有正定解和幂零解的充要条件

给出了矩阵方程AX=A+X有解、实对称解、正定解和幂零解的充要条件.     

非线性耦合标量场方程显式解析解的研究

利用两种不同的变换,获得了一类非线性耦合标量场方程的若干类型的精确解析解,其中包括孤子解、奇性孤波解和三角函数解,从而丰富了方程解的内容。这些结论可以应用于其它的非线性方程。此外还纠正了一些文献的部分结论。     

非线性演化方程显式精确解的新算法

本文给出了一种求解非线性演化方程的新算法 .将这种算法运用于变形浅水波方程 ,获得了八组显式精确解 ,其中包括新的孤波解和周期解 .借助于 Mathematica软件 ,这种算法能够在 Computer上实现 .     

Bifurcations of travelling wave solutions for a two-component camassa-holm equation

By using the method of dynamical systems to the two-component generalization of the Camassa-Holm equation, the existence of solitary wave solutions, kink and anti-kink wave solutions, and uncountably infinite many breaking wave solutions, smooth and non-smooth periodic wave solutions is obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. Some exact explicit parametric representations of travelling wave solutions are listed.     

Exact traveling wave solutions and bifurcations for the Dullin-Gottwald-Holm equation

Utilizing the methods of dynamical system theory, the Dullin-Gottwald-Holm equation is studied in this paper. The dynamical behaviors of the traveling wave solutions and their bifurcations are presented in different parameter regions. Furthermore, the exact explicit forms of all possible bounded solutions, such as solitary wave solutions, periodic wave solutions and breaking loop wave solutions are obtained.     

Vortical and self-similar flows of 2D compressible Euler equations

This paper presents the vortical and self-similar solutions for 2D compressible Euler equations using the separation method. These solutions complement Makino’s solutions in radial symmetry without rotation. The rotational solutions provide new information that furthers our understanding of ocean vortices and reference examples for numerical methods. In addition, the corresponding blowup, time-periodic or global existence conditions are classified through an analysis of the new Emden equation. A conjecture regarding rotational solutions in 3D is also made.     

Interaction solutions to Hirota-Satsuma-Ito equation in (2+1)-dimensions

Abundant exact interaction solutions, including lump-soliton, lump-kink, and lump-periodic solutions, are computed for the Hirota-Satsuma-Ito equation in (2+1)-dimensions, through conducting symbolic computations with Maple. The basic starting point is a Hirota bilinear form of the Hirota-Satsuma-Ito equation. A few three-dimensional plots and contour plots of three special presented solutions are made to shed light on the characteristic of interaction solutions.     

二维RLW方程和二维SRLW方程的显式精确解

本文讨论了二维RLW方程和二维SRLW方程孤立波解的性态，通过直接积分的方法求出了这两个方程的显式精确孤立波解，并通过选取初始条件的方法求出了二维RLW方程和二维SRLW方程的另一类精确行波解．     

多目标规划强较多有效解和弱较多有效解的标量化和对偶性

利用n维Euclid空间中较多锥的闭包,定义了多目标规划强较多有效解.利用较多锥闭包的Minkowski泛函,给出了强较多有效解和弱较多有效解的标量化结果,并讨论了强较多有效解和弱较多有效解的标量对偶问题及其应用.     

一般Lidstone边值问题的n个正解的存在性

考察了含所有偶数阶导数的一般Lidstone边值问题的正解和对称正解.通过 选择合适的Banach空间和锥,对该问题建立了n个正解或者对称正解的存在性,其 中n是一个任意的自然数.基本工具是等价范数和锥拉伸与锥压缩型的Krasnosel'skii 不动点定理. 结论的主要条件是局部的.换言之,如果非线性项f在某些有界集上的 "高度"是适当的,则该问题可以具有n个正解.     

一类非线性演化方程新的显式行波解

借助Mathematica软件和吴方法,采用双曲函数法,获得了一类非线性演化方程u_tt+au_xx+bu+cu2+du3=0的多组行波解,其中包括周期解与孤子解.这种方法也适用于其他非线性方程或方程组.     

Convex solutions of a general similarity boundary layer equation for power-law fluids

This paper is devoted to a general similarity boundary layer equation for power-law fluids, which includes many important similarity boundary layer problems such as the Falker-Skan equation and the magnetohydrodynamic boundary layer equation which arises in the study of self-similar solutions of the two-dimensional steady laminar boundary layer flow for an incompressible electrically conducting power-law fluids along an isolated surface in the presence of an exterior magnetic field orthogonal to the flow. By a rigorous mathematical analysis, the uniqueness, existence and nonexistence results for convex solutions, normal convex solutions and generalized convex solutions to the general similarity boundary layer equation are established. Also the asymptotic behavior of the normal convex solutions at the infinity are displayed.     

The improved Fan sub-equation method and its application to the SK equation

In this paper, many types of exact solutions of a first-order nonlinear ordinary differential equation called Fan sub-equation, is further investigated by using bifurcation method of dynamical systems. As a result, more types of exact solutions to Sawada-Kotera (SK) equation are obtained, which include more general soliton solutions, kink solutions, triangular function solutions, Jacobian elliptic function solutions with double periods and so on.     

一类时滞反应扩散方程的周期解和概周期解

通过构造上、下控制函数，结合上、下解方法及相应的单调迭代方法研究了一类时滞反应扩散方程，证明了在反应项非单调时，如果一雏边值问题存在一对周期（或概周期）上、下解，则方程一定存在唯一的周期（或概周期）解．并给出了二维边值问题周期（或概周期）解存在唯一性的充分条件．推广了已有的一些结果。     

Some results on difference Riccati equations

Some properties of solutions for the difference Riccati equations are obtained. The existence and forms of rational solutions, and the Borel exceptional value, zeros, poles and fixed points of transcendental solutions are researched.     

Two-dimensional steady-state general solution for isotropic thermoelastic materials with applications. I: General solutions and fundamental solutions

Four steady-state general solutions are derived in this paper for the two-dimensional equation of isotropic thermoelastic materials. Using the differential operator theory, three general solutions can be derived and expressed in terms of one function, which satisfies a six-order partial differential equation. By virtue of the Almansi’s theorem, the three general solutions can be transferred to three general solutions which are expressed in terms of two harmonic functions, respectively. At last, a integrate general solution expressed in three harmonic functions is obtained by superposing the obtained two general solutions. The proposed general solution is very simple in form and can be used easily in certain boundary problems. As two examples, the fundamental solutions for both a line heat source in the interior of infinite plane and a line heat source on the surface of semi-infinite plane are presented by virtue of the obtained general solutions.     

双函数法及一类非线性发展方程的精确行波解

给出一种求解非线性发展方程精确行波解的新方法：双函数法。使用此方法，获得了一类非线性发展方程的许多精确行波解，其中包括孤波解和周期解，推广了文献用其它方法取得的结果，同时还获得了许多新的弧波解和周期解，借助于Mathemat-ica，此方法能部分地在计算机上实现。