应用辅助方程法求Zakharov方程的精确解

应用辅助方程法求得Zakharov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,可得到孤立波解:当对三角函数解中的参数取特殊值时,可得到周期波函数解.实践表明:辅助方程法在非线性光学、量子光学、激光物理和等离子体物理等领域具有广泛的应用.     

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.     

应用G/G′展开法求Ostrovsky方程的孤子解

应用(G/G')展开法构造出(1+1)维0strovsky方程的10组精确解,这些解的类型主要包含双曲函数通解、三角函数通解和有理函数通解三种形式.对解的性质进行了相应地分析,当对双曲函数通解中的参数取特殊值时,可以得到了孤立波解.当对三角函数通解中引中的参数取特殊值时,可以得到对应的周期波函数解.     

一类5阶KdV方程的孤立波解

近年来,关于3阶KdV方程的孤立波解得到迅速发展,而对于5阶KdV方程的孤立波解文献报道较少.本文主要采用Sine-Cosine展开法得到了一类5阶KdV方程的孤立波解;然后利用Matlab计算软件,获得了孤立波解的图形,其结果展示了孤立子与系数之间的相互关系;最后,应用所得的结果分别得到了Lax方程, SK方程, CDG方程的孤立波解.     

Dodd-Bullough-Mikhailov方程的精确解

利用(G'/G)法求解了Dodd-Bullough-Mikhailov的精确解,得到了Dodd-Bullough-Mikhailov方程的用双曲函数,三角函数和有理函数表示的三类精确行波解.由于方法中的G为某个二阶常系数线性ODE的通解,故方法具有直接、简洁的优点;更重要的是,方法可用于求得其它许多非线性演化方程的行波解.如果对其中双曲函数表示的行波解中的参数取特殊值,那么可得已有的孤波解.     

随机广义KdV方程的随机精确解

通过一个辅助性方程和埃米尔特变换研究广义随机KdV方程的随机雅克比椭圆函数类波解.此外,还通过椭圆函数在模数取极限m→0和m→1的情况,给出方程的随机类孤子解和随机三角函数波解,所得结果丰富了广义随机KdV方程的精确解.     

带色散项的高阶非线性Schrdinger方程的精确解

对一类带色散项的高阶非线性Schrdinger方程的精确解进行研究.通过行波约化,将一类带色散项的高阶非线性Schrdinger方程化为一个高阶非线性常微分方程.再借助于计算机代数系统Mathematica通过构造非线性常微分方程的精确解,成功获得了一系列含有多个参数的包络型精确解,当精确解中参数取特殊值时可以得到两种新型的复合孤子解.并讨论了这两种孤子解存在的参数条件.     

几个非线性发展方程的精确解

应用改进F/G展开法求得(2+1)维BBM方程、(1+1)维Benjiamin Ono方程、广义(2+1)维ZK-MEW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,可得到孤立波解:当对三角函数解中的参数取特殊值时,可得到周期波函数解.实践表明,F/G展开法在非线性发展方程中具有广泛的应用.     

变系数李方程组的精确行波解

利用修正的简单方程法对变系数李方程组进行求解,给出了变系数李方程组的双曲函数形式的行波解,当参数取特殊值时,便可以得到该方程组的精确孤波解.     

广义耦合KdV孤子方程的达布变换及其偶孤子解

根据广义耦合KdV孤子方程的Lax对, 借助谱问题的规范变换, 一个包含多参数的达布变换被构造出来. 利用达布变换来产生广义耦合KdV孤子方程的偶孤子解, 并且用行列式的形式来表达广义耦合KdV孤子方程的偶孤子解. 作为应用, 广义耦合KdV孤子方程的偶孤子解的前两个例子被给出.     

Unique solutions

It is folklore that if a continuous function on a complete metric space has approximate roots and in a uniform manner at most one root, then it actually has a root, which of course is uniquely determined. Also in Bishop's constructive mathematics with countable choice, the general setting of the present note, there is a simple method to validate this heuristic principle. The unique solution even becomes a continuous function in the parameters by a mild modification of the uniqueness hypothesis. Moreover, Brouwer's fan theorem for decidable bars turns out to be equivalent to the statement that, for uniformly continuous functions on a compact metric space, the crucial uniform “at most one” condition follows from its non‐uniform counterpart. This classification in the spirit of the constructive reverse mathematics, as propagated by Ishihara and others, sharpens an earlier result obtained jointly with Berger and Bridges. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Blow-up solutions and global solutions for nonlinear parabolic problems

This paper deals with the blow-up of positive solutions for a nonlinear parabolic equation subject to nonlinear boundary conditions. We obtain the conditions under which the solutions may exist globally or blow up in a finite time, by a new approach. Moreover, upper estimates of the “blow-up time”, blow-up rate and global solutions are obtained also.     

Viscosity solutions, almost everywhere solutions and explicit formulas

Consider the differential inclusion in . We exhibit an explicit solution that we call fundamental. It also turns out to be a viscosity solution when properly defining this notion. Finally, we consider a Dirichlet problem associated to the differential inclusion and we give an iterative procedure for finding a solution.

     

MAJOR－EFFICIENT SOLUTIONS AND WEAKLY MAJOR－EFFICIENT SOLUTIONS OF MULTIOBJECTIVE PROGRAMMING

ＭＡＪＯＲ－ＥＦＦＩＣＩＥＮＴＳＯＬＵＴＩＯＮＳＡＮＤＷＥＡＫＬＹＭＡＪＯＲ－ＥＦＦＩＣＩＥＮＴＳＯＬＵＴＩＯＮＳＯＦＭＵＬＴＩＯＢＪＥＣＴＩＶＥＰＲＯＧＲＡＭＭＩＮＧ￥ＨＵＹＵＤＡ（Ｄｅｐｔ．ｏｆＡｐｐｌ．Ｍａｔｈ．，ＳｈａｎｇｈａｉＪｉａｏＴｏｎ...     

Blow-up solutions and global solutions for nonlinear parabolic problems

This paper deals with the blow-up of positive solutions for a nonlinear parabolic equation subject to nonlinear boundary conditions. We obtain the conditions under which the solutions may exist globally or blow up in a finite time, by a new approach. Moreover, upper estimates of the “blow-up time”, blow-up rate and global solutions are obtained also.     

Global solutions and self-similar solutions of semilinear wave equation

We prove existence, uniqueness and regularity results for the global solutions of the semilinear wave equations. In particular, we show existence of regular self-similar solutions. Also, we build some finite-energy asymptotically self-similar solutions. Received: 20 September 1999; in final form: 10 May 2000 / Published online: 25 June 2001     

Energy solutions for polymer aqueous solutions in two dimension

The aim of this article is to study a nonlinear system modeling a Non-Newtonian fluid of polymer aqueous solutions. We are interested here in the existence of weak solutions for the stationary problem in a bounded plane domain or in two-dimensional exterior domain. Due to the third order of derivatives in the non-linear term, it’s difficult to obtain solutions satisfying energy inequality. But with a good choice of boundary conditions, an adapted special basis and the use of the good properties of the trilinear form associated to the non-linear term, we obtain energy solutions. The problem in bounded domains is treated and the more difficult problem on non bounded domains too.     

Non-monotonic solutions and continuously differentiable solutions of conjugacy equations

In this paper solutions of conjugacy equation φ(f(x))=g(φ(x)) for a strictly decreasing continuous given function f and a continuous given function g (maybe non-monotonic) are constructed by piecewise defining. We determine the conditions for piecewise continuously differentiable solutions of conjugacy equations with a strictly decreasing continuously differentiable given function f and a continuously differentiable given function g. Finally, the recursive algorithm is implemented in MATLAB software and two examples are respectively presented for a non-monotonic solution and a continuously differentiable one.