Gauge-invariant description of several (2+1)-dimensional integrable nonlinear evolution equations

We obtain new gauge-invariant forms of two-dimensional integrable systems of nonlinear equations: the Sawada-Kotera and Kaup-Kuperschmidt system, the generalized system of dispersive long waves, and the Nizhnik-Veselov-Novikov system. We show how these forms imply both new and well-known twodimensional integrable nonlinear equations: the Sawada-Kotera equation, Kaup-Kuperschmidt equation, dispersive long-wave system, Nizhnik-Veselov-Novikov equation, and modified Nizhnik-Veselov-Novikov equation. We consider Miura-type transformations between nonlinear equations in different gauges. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 160, No. 1, pp. 35–48, July, 2009.     

Comment on “New types of exact solutions for nonlinear Schrodinger equation with cubic nonlinearity”

In this comment we analyze the paper [Abdelhalim Ebaid, S.M. Khaled, New types of exact solutions for nonlinear Schrodinger equation with cubic nonlinearity, J. Comput. Appl. Math. 235 (2011) 1984-1992]. Using the traveling wave, Ebaid and Khaled have found “new types of exact solutions for nonlinear Schrodinger equation with cubic nonlinearity”. We demonstrate that the authors studied the well-known nonlinear ordinary differential equation with the well-known general solution. We illustrate that Ebaid and Khaled have looked for some exact solution for the reduction of the nonlinear Schrodinger equation taking the general solution of the same equation into account.     

A sub-ODE method for generalized Gardner and BBM equation with nonlinear terms of any order

In this paper, a method with the aid of a sub-ODE and its solutions is used for constructing new periodic wave solutions for nonlinear Gardner equation and BBM equation with nonlinear terms of any order arising in mathematical physics. As a result, many exact traveling wave solutions are successfully obtained. The method in the paper is very direct and it can also be applied to other nonlinear evolution equations.     

Methods of judging shape of solitary wave and solution formulae for some evolution equations with nonlinear terms of high order

In this paper, we present several methods of judging shape of the solitary wave and solution formulae for some nonlinear evolution equations by means of Lienard equations. Then, using the judgement methods and solution formulae, we obtain solutions of the solitary wave for some of important nonlinear evolution equations, which include generalized modified Boussinesq, generalized nonlinear wave, generalized Fisher, generalized Klein-Gordon and generalized Zakharov equations. Some new solitary-wave solutions are found for the equations.     

Nonlinear evolution equations for describing waves in bubbly liquids with viscosity and heat transfer consideration

Nonlinear evolution equations of the fourth order and its partial cases are derived for describing nonlinear pressure waves in a mixture liquid and gas bubbles. Influence of viscosity and heat transfer is taken into account. Exact solutions of nonlinear evolution equation of the fourth order are found by means of the simplest equation method. Properties of nonlinear waves in a liquid with gas bubbles are discussed.     

Applications of He’s principles to partial differential equations

This paper applies the variational iteration method (VIM) and semi-inverse variational principle to obtain solutions of linear and nonlinear partial differential equations. The nonlinear model is considered from gas dynamics, fluid dynamics and Burgers equation. The linear model is the heat transfer (diffusion) equation. Results show that variational iteration method is a powerful mathematical tool for solving linear and nonlinear partial differential equations, and therefore, can be widely applied to engineering problems.     

Local Discontinuous Galerkin Methods for Three Classes of Nonlinear Wave Equations

In this paper, we further develop the local discontinuous Galerkin method to solve three classes of nonlinear wave equations formulated by the general KdV-Burgers type equations, the general fifth-order KdV type equations and the fully nonlinear K(n, n, n) equations, and prove their stability for these general classes of nonlinear equations. The schemes we present extend the previous work of Yan and Shu [30, 31] and of Levy, Shu and Yan [24] on local discontinuous Galerkin method solving partial differential equations with higher spatial derivatives. Numerical examples for nonlinear problems are shown to illustrate the accuracy and capability of the methods. The numerical experiments include stationary solitons, soliton interactions and oscillatory solitary wave solutions.The numerical experiments also include the compacton solutions of a generalized fifthorder KdV equation in which the highest order derivative term is nonlinear and the fully nonlinear K (n, n, n) equations.     

几种广义非线性发展方程的新解

本文研究了构造了广义Kd V方程和广义KP-Burgers方程等几种广义非线性发展方程的新解的问题.利用三种辅助方程及其新解,获得了广义Kd V方程和广义KP-Burgers方程等几种广义非线性发展方程的新解.这些解由双曲余割函数、双曲正切函数、双曲正割函数、双曲余切函数和余割函数组成.     

A variational approach to stochastic nonlinear parabolic problems

This work is concerned with an optimal control approach to stochastic nonlinear parabolic diffusion equations with monotonically increasing nonlinearity. This approach leads to sharper existence and uniqueness results under minimal growth conditions on nonlinear diffusion coefficients.     

The variational iteration method for analytic treatment for linear and nonlinear ODEs

In this work, the variational iteration method (VIM) is used for analytic treatment of the linear and nonlinear ordinary differential equations, homogeneous or inhomogeneous. The method is capable of reducing the size of calculations and handles both linear and nonlinear equations, homogeneous or inhomogeneous, in a direct manner. However, for concrete problems, a huge number of iterations are needed for a reasonable level of accuracy.     

Nonautonomous first integrals and general solutions of the KdV‐Burgers and mKdV‐Burgers equations with the source

The method for constructing first integrals and general solutions of nonlinear ordinary differential equations is presented. The method is based on index accounting of the Fuchs indices, which appeared during the Painlevé test of a nonlinear differential equation. The Fuchs indices indicate us the leading members of the first integrals for the origin differential equation. Taking into account the values of the Fuchs indices, we can construct the auxiliary equation, which allows to look for the first integrals of nonlinear differential equations. The method is used to obtain the first integrals and general solutions of the KdV‐Burgers and the mKdV‐Burgers equations with a source. The nonautonomous first integrals in the polynomials form are found. The general solutions of these nonlinear differential equations under at some additional conditions on the parameters of differential equations are also obtained. Illustrations of some solutions of the KdV‐Burgers and the mKdV‐Burgers are given.     

Blow up of solutions of pseudoparabolic equations

We obtain sufficient conditions for the blow up of solutions of the initial-boundary value problem for nonlinear pseudoparabolic equation involving nonlinear convective term.     

一类非线性色散型发展方程反问题的整体解

将一类非线性色散型发展方程反问题转化为抽象空间非线性发展方程Cauchy问题。利用半群方法和赋等价范数技巧,建立了该类抽象发展方程整体解的存在唯一性定理,并应用于所论反问题,得到了该类非线性色散型发展方程反问题整体解的存在唯一性定理,本质地改进了袁忠信得出的解的局部存在唯一性结果。     

一类三阶非线性微分方程解的不稳定性*

文献[1]讨论了非线性缓变系统的渐近稳定性,文献[2]讨论了三阶变系数线性微分方程解的不稳定性。本文应用文献[1]、[2]的方法讨论一类三阶非线性微分方程解的不稳定性。     

The extended Riccati equation mapping method for variable-coefficient diffusion-reaction and mKdV equations

In this paper, the extended Riccati equation mapping method is proposed to seek exact solutions of variable-coefficient nonlinear evolution equations. Being concise and straightforward, this method is applied to certain type of variable-coefficient diffusion-reaction equation and variable-coefficient mKdV equation. By means of this method, hyperbolic function solutions and trigonometric function solutions are obtained with the aid of symbolic computation. It is shown that the proposed method is effective, direct and can be used for many other variable-coefficient nonlinear evolution equations.     

The Davey–Stewartson equation with sources derived from nonlinear variable separation method

In this paper, by taking the Davey–Stewartson equation as an example, a new construction procedure based on nonlinear variable separation method is presented to obtain nonlinear evolution equations with sources for the first time.     

非线性演化方程的孤立波解

用齐次平衡原则和辅助微分方程方法得到了6个重要的n次非线性演化方程的孤立波解.辅助微分方程方法的主要思想是借助简单的可解微分方程的解去构造复杂的非线性演化方程的行进波解.这里简单的可解微分方程称为辅助微分方程.本文使用的辅助方程有双曲正割幂型解或双曲正切幂型解.     

Nonclassical symmetries of a class of nonlinear partial differential equations with arbitrary order and compatibility

In this paper, we show that for a class of nonlinear partial differential equations with arbitrary order the determining equations for the nonclassical reduction can be obtained by requiring the compatibility between the original equation and the invariant surface condition. The nonlinear wave equation and the Boussinesq equation all serve as examples illustrating this fact.     

Generalized Extended Tanh-Function Method for Traveling Wave Solutions of Nonlinear Physical Equations

In this paper, the generalized extended tanh-function method is used for constructing the traveling wave solutions of nonlinear evolution equations. We choose Fisher's equation, the nonlinear schrödinger equation to illustrate the validity and advantages of the method. Many new and more general traveling wave solutions are obtained. Furthermore, this method can also be applied to other nonlinear equations in physics.     

一类带强色散项DGH方程的精确解

利用改进的F-展开法,求出了一类带强色散项DGH方程的一系列类孤子解,三角函数周期解和有理数解,方程结合了KdV方程的线性色散项和C-H方程的非线性色散项.而且改进的F-展开法在借助于计算机符号系统Mathematica(Maple)下,操作方便,适用于大量的非线性偏微分方程(组),并有助于发现新解.