On the Marchenko System and the Long-time Behavior of Multi-soliton Solutions of the One-dimensional Gross-Pitaevskii Equation

We establish a rigorous well-posedness results for the Marchenko system associated to the scattering theory of the one dimensional Gross-Pitaevskii equation (GP). Under some assumptions on the scattering data, these well-posedness results provide regular solutions for (GP). We also construct particular solutions, called Nsoliton solutions as an approximate superposition of traveling waves. A study for the asymptotic behaviors of such solutions when t → ± ∞ is also made.     

Generalization of the simplest equation method for nonlinear non-autonomous differential equations

It is known that the simplest equation method is applied for finding exact solutions of autonomous nonlinear differential equations. In this paper we extend this method for finding exact solutions of non-autonomous nonlinear differential equations (DEs). We applied the generalized approach to look for exact special solutions of three Painlevé equations. As ODE of lower order than Painlevé equations the Riccati equation is taken. The obtained exact special solutions are expressed in terms of the special functions defined by linear ODEs of the second order.     

应用改进的简单方程法求非线性数学物理方程的精确解

应用改进的简单方程法求得Cahn-Allen方程和Jimbo-Miwa方程的精确解,这些解包括双曲函数解、三角函数解.当对双曲函数解中的参数取特殊值时,可以得到了孤立波解.当对三角函数解中的参数取特殊值时,可以得到对应的周期波函数解.实践证明,简单方程法对于研究非线性数学物理方程具有非常广泛的应用意义.     

On reduction of some classes of partial differential equations to equations with fewer variables and exact solutions

We establish a connection between the fundamental solutions to some classes of linear nonstationary partial differential equations and the fundamental solutions to other nonstationary equations with fewer variables. In particular, reduction enables us to obtain exact formulas for the fundamental solutions of some spatial nonstationary equations of mathematical physics (for example, the Kadomtsev-Petviashvili equation, the Kelvin-Voigt equation, etc.) from the available fundamental solutions to one-dimensional stationary equations.     

Quasi-rational function solutions of an elliptic equation and its application to solve some nonlinear evolution equations

Using the differential transformation method and the homogeneous balance method, some new solutions of an auxiliary elliptic equation are obtained. These solutions possess the forms of rational functions in terms of trigonometric functions, hyperbolic functions, exponential functions, power functions, elliptic functions and their operation and composite functions and so on, which are so-called quasi-rational function solutions. Based on these new quasi-rational functions solutions, a direct method is proposed to construct the exact solutions of some nonlinear evolution equations with the aid of symbolic computation. The coupled KdV-mKdV equation and Broer-Kaup equations are chosen to illustrate the effectiveness and convenience of the suggested method for obtaining quasi-rational function solutions of nonlinear evolution equations.     

On the sets of exact and approximate solutions

We consider the problem how big is the set of solutions of a given functional equation in the set of approximate solutions. It happens that in the cases of linear functional equations (like Cauchy, Jensen) or linear inequalities (like convex, Jensen convex) the sets of solutions are very small subsets of the sets of approximate solutions. The situation is different in the cases of superstable equations (like exponential or d'Alembert).     

A series of new exact solutions of nonlinear evolution equations

With the aid of computer symbolic computation system Maple, the generalized auxiliary equation method is first applied to two nonlinear evolution equations, namely, the nonlinear elastic rod equation and (2 + 1)‐dimensional Boiti‐Leon‐Pempinelli equation. As a results, some new types of exact traveling wave solutions are obtained which include bell and kink profile solitary wave solutions, and triangular periodic wave solutions and singular solutions. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in mathematical physics. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

On Nonlinear Hyperbolic Functional Differential Equations

In this paper, existence of weak solutions of second order evolution equations is proved and some properties of the solutions are shown. The results are applied to higher order nonlinear hyperbolic functional differential equations.     

Remarks on Third-Order ODEs Relevant to the Kuramoto-Sivashinsky Equation

We are interested in the third-order ordinary differential equations (ODEs) which are related to the Kuramoto-Sivashinsky equation. So-called steady solutions of the Kuramoto-Sivashinsky equation are known to admit several types; for example, bounded global solutions or periodic solutions. We show that, in addition to these, there exist solutions which blow up on bounded intervals. Moreover, for certain classes of these ODEs, the nonexistence of nontrivial bounded entire solutions is exhibited.     

一类反应扩散方程解的熄灭现象

利用能量估计方法讨论了下述反应扩散方程的初边值问题解的渐近性态,分别给出解熄灭的充分条件和必要条件。这里λ>0,γ>0,β>0为常数,Ω?RN为有界域。文末给出说明文中方法处理高阶方程的例子。     

Analytic and approximate solutions of the space- and time-fractional telegraph equations

The Adomian decomposition method is used to obtain analytic and approximate solutions of the space-and time-fractional telegraph equations. The space- and time-fractional derivatives are considered in the Caputo sense. The analytic solutions are calculated in the form of series with easily computable terms. Some examples are given. The results reveal that the Adomian method is very effective and convenient.     

Oscillatory properties for semilinear degenerate hyperbolic equations of second order

We consider three kind of oscillatory properties of the solutions to semilinear degenerate hyperbolic equations. Several sufficient conditions for the oscillation or non-oscillation are presented. In particular, they give us the positivity of the solutions for semilinear hyperbolic equations degenerating at initial point in one space dimension. Moreover we establish a few oscillatory conditions for the solutions of the mixed problem reduced to in one space dimension.     

The extended Riccati equation method for travelling wave solutions of ZK equation

In this article, the extended Riccati equation method is applied to seeking more general exact travelling wave solutions of the ZK equation. The traveling wave solutions are expressed by the hyperbolic functions, the trigonometric functions and the rational functions. When the parameters are taken as special values, the solitary wave solutions are obtained from the hyperbolic function solutions. Similarly, the periodic wave solutions are also obtained from the trigonometric function solutions. The approach developed in this paper is effective and it may also be used for solving many other nonlinear evolution equations in mathematical physics.     

一个组合方程的单孤子解和周期尖波解

构造一个组合方程的单孤子解和周期尖波解.应用格林函数的性质,以及求一个非线性偏微分方程(简称PDE)弱解的方法.求出了这个组合方程的单孤子解和周期尖波解,推广了前人的研究成果.     

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.     

A blow-up theorem for a discrete semilinear wave equation

In this paper, a discretization of a semilinear wave equation whose nonlinear term is a power function is investigated. It is shown that, when a condition on the initial value problem, similar to that governing the existence of blow-up solutions for the original continuous equation is met, the newly introduced difference equation has blow-up solutions with characteristics corresponding to those of the blow-up solutions for the original equation.     

GLOBAL WEAK SOLUTIONS OF THE RELATIVISTIC VLASOV-KLEIN-GORDON SYSTEM IN TWO DIMENSIONS

In this paper we consider the system of classical particles coupled with a Klein-Gordon field in two dimensions.We establish a-priori-bounds on the solutions of this system with initial data satisfying a size restriction derived from conservation of energy.This result,together with the smoothing of"velocity averaging",yields the existence of global weak solutions to the corresponding restricted initial value problem.The size restriction is necessary since energy of the system is indefinite.Finally,we show that the weak solutions preserve the total mass.     

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.     

New traveling waves solutions to generalized Kaup-Kupershmidt and Ito equations

In this paper we consider a special fifth-order KdV equation with constant coefficients and we obtain traveling wave solutions for it, using the projective Riccati equation method. By mean of a scaling, exact solutions to general Kaup-Kupershmidt (KK) equation are obtained. As a particular case, exact solutions to standard KK equation can be derived. Using the same method, we obtain exact solutions to standard Ito equation. By mean of scaling, new exact solutions to general Ito equation are formally derived.     

Euler integral symmetries for a deformed Heun equation and symmetries of the Painlevé PVI equation

Euler integral transformations relate solutions of ordinary linear differential equations and generate integral representations of the solutions in a number of cases or relations between solutions of constrained equations (Euler symmetries) in some other cases. These relations lead to the corresponding symmetries of the monodromy matrices. We discuss Euler symmetries in the case of the simplest Fuchsian system that is equivalent to a deformed Heun equation, which is in turn related to the Painlevé PVI equation. The existence of integral symmetries of the deformed Heun equation leads to the corresponding symmetries of the PVI equation. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 2, pp. 252–264, May, 2008.