Exact solutions of the Kudryashov-Sinelshchikov equation

The Kudryashov-Sinelshchikov equation for describing the pressure waves in liquid with gas bubbles is studied. New exact solutions of this equation are found. Modification of truncated expansion method is used for obtaining exact solution of this equation.     

Exact Solitary-wave Solutions and Periodic Wave Solutions for Generalized Modified Boussinesq Equation and the Effect of Wave Velocity on Wave Shape

By means of the undetermined assumption method, we obtain some new exact solitary-wave solutions with hyperbolic secant function fractional form and periodic wave solutions with cosine function form for the generalized modified Boussinesq equation. We also discuss the boundedness of these solutions. More over, we study the correlative characteristic of the solitary-wave solutions and the periodic wave solutions along with the travelling wave velocity＇s variation.     

Some exact Wick type stochastic generalized Boussinesq equation solutions

The resolution of the stochastic generalized Boussinesq equation driven by a white noise is undertaken. Explicit solutions are found thanks to a white noise functional approach and the F-expansion method. Among these solutions, periodic and solitonic ones are pointed out.     

Trigonometric and hyperbolic type solutions to a generalized Drinfel’d-Sokolov equation

A class of trigonometric and hyperbolic type solutions to the generalized Drinfel’d-Sokolov (GDS) equations

     

Solitary and periodic solutions of the generalized Kuramoto-Sivashinsky equation

The generalized Kuramoto-Sivashinsky equation in the case of the power nonlinearity with arbitrary degree is considered. New exact solutions of this equation are presented.      

一般变系数KdV方程的精确解

By asing the nonclassical method of symmetry reductions, the exact solutions for general variable-coefficient KdV equation with dissipative loss and nonuniformity terms are obtained. When the dissipative loss and nonuniformity terms don‘t exist, the multisoliton solutions are found and the corresponding Painleve II type equation for the variable-coefficient KdV equation is given.     

Sufficient close-to-necessary conditions for the blowup of solutions to a strongly nonlinear generalized Boussinesq equation

An initial boundary value problem for the generalized Boussinesq equation with allowance for linear dissipation and free electron sources is considered. The strong generalized time-local solvability of the problem is proved. Sufficient conditions are obtained for the blowup of the solution and for time-global solvability. Two-sided estimates of the blowup time are derived.     

一个曲面波方程的精确解

Abstract. Some exact travelling wave solutions and rational travelling wave solutions of a sur-face wave equation in a convecting fluid are given in this paper.     

Exact travelling wave solutions of the generalized Bretherton equation

We search for exact travelling wave solutions of the generalized Bretherton equation for integer, greater than one, values of the exponent m of the nonlinear term by two methods: the truncated Painlevé expansion method and an algebraic method. We find periodic solutions for m=2 and m=5, to add to those already known for m=3; in all three cases these solutions exist for finite intervals of the wave velocity. We also find a “kink” shaped solitary wave for m=5 and a family of elementary unbounded solutions for arbitrary m.     

Exact and explicit solutions for a nonlinear extended iterative differential equation

This paper is concerned with a nonlinear iterative functional differential equation x′(z) = 1/x(p(z) + bx′(z)). By constructing a convergent power series solution of an auxiliary equation, analytic solutions of the original equation are obtained. We discuss not only in the general case, but also in critical cases, especially for α given in Schröder transformation is a root of the unity. And in case (H4), we dealt with the equation under the Brjuno condition, which is weaker than the Diophantine condition. Moreover, the exact and explicit solution of the original equation has been investigated for the first time. Such equations are important in both applications and the theory of iterations.     

New solutions of the Schwarzian Korteweg-de Vries equation in 2+1 dimensions based on weak symmetries

We consider the (2+1)-dimensional integrable Schwarzian Korteweg-de Vries equation. Using weak symmetries, we obtain a system of partial differential equations in 1+1 dimensions. Further reductions lead to second-order ordinary differential equations that provide new solutions expressible in terms of known functions. These solutions depend on two arbitrary functions and one arbitrary solution of the Riemann wave equation and cannot be obtained by classical or nonclassical symmetries. Some of the obtained solutions of the Schwarzian Korteweg-de Vries equation exhibit a wide variety of qualitative behaviors; traveling waves and soliton solutions are among the most interesting. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 3, pp. 380–390, June, 2007.     

扰动Boussinesq方程的近似守恒律

构造了具有扰动项的Boussinesq方程的近似守恒向量和近似守恒律.在方程允许拉格朗日函数的情况下,利用欧拉方程的部分拉格朗日函数方法,研究了含有一阶线性组合扰动项的Boussineq方程的近似守恒律.给出了该方程的近似守恒向量及近似守恒律的分类结果.     

New exact solutions of a generalised Boussinesq equation with damping term and a system of variant Boussinesq equations via double reduction theory

The conservation laws of a generalised Boussinesq (GB) equation with damping term are derived via the partial Noether approach. The derived conserved vectors are adjusted to satisfy the divergence condition. We use the definition of the association of symmetries of partial differential equations with conservation laws and the relationship between symmetries and conservation laws to find a double reduction of the equation. As a result, several new exact solutions are obtained. A similar analysis is performed for a system of variant Boussinesq (VB) equations.     

New solutions for the modified generalized Degasperis-Procesi equation

In this paper, using three distinct computational methods we obtain some new exact solutions for the generalized modified Degasperis-Procesi equation (mDP equation) u_t-u_xxt+(b+1)u²u_x=bu_xu_xx+uu_xxx. We show the graph of some of the new solutions obtained here with the aim to illustrate their physical relevance. Mathematica is used. Finally some conclusions are given.     

Similarity reductions and similarity solutions for the generalized diffusion equation

Herein, the generalized diffusion equation that encompasses the nonlinear diffusion equation with a source term and the Boussinesq equation in hydrology as its particular form and appears in a wide variety of physical and engineering applications has been analyzed via symmetry method that was developed by Steinberg. According to physical situations, in each case, the similarity variables obtained have led us to an ordinary differential equation, and we acquire some new solutions by solving the ODEs. Copyright © 2015 John Wiley & Sons, Ltd     

一类四阶偏微分方程的对称分析及级数解

研究了一类四阶偏微分方程的李对称,构造了方程所容许的李对称的优化系统,进行了对称约化,得到了精确解.进一步,基于幂级数理论,得到了这类四阶偏微分方程的幂级数解.     

Blowup of solutions for improved Boussinesq type equation

The paper studies the existence and uniqueness of local solutions and the blowup of solutions to the initial boundary value problem for improved Boussinesq type equation u_tt−u_xx−u_xxtt=σ(u)_xx. By a Galerkin approximation scheme combined with the continuation of solutions step by step and the Fourier transform method, it proves that under rather mild conditions on initial data, the above-mentioned problem admits a unique generalized solution u∈W^2,∞([0,T];H²(0,1)) as long as . In particular, when σ(s)=as^p, where a≠0 is a real number and p>1 is an integer, specially a<0 if p is an odd number, the solution blows up in finite time. Moreover, two examples of blowup are obtained numerically.