非线性偏微分方程的精确行波解

Exact Traveling Wave Solutions of Nonlinear Partial Differential Equations

计算机科学是研究信息与计算的理论基础以及它们在计算机系统中如何实现和应用的学科。非线性科学是研究自然界中非线性问题的共性。计算机代数系统出现后,人们开始用计算机进行符号计算和自动推理,科学家们对非线性问题的研究也一直与计算机科学的发展息息相关。在数学和物理学领域,人们习惯用方程来描述客观事物的运动规律。非线性偏微分方程是可以描述客观事物非线性演化过程的数学物理方程,但是数学上几乎不存在通用的方法求解此类方程。文中采用辅助函数法求解非线性Benjamin-Bona-Mahonye方程的精确行波解。借助符号计算系统Maple,通过修正辅助函数法中参数m的取值范围,获得并比较了方程行波解在数量和形式上的变化。研究结果表明,修正后的辅助函数法能够获得形式更为丰富的行波解。

Computer science is a discipline that studying theoretical basis of information and computation and how they are implemented and applied in computer systems.Non-linear science studies the commonness of nonlinear problems in nature.After the emergence of computer algebra system,people began to use the computer for symbol calculation and automatic reasoning,and scientists study of nonlinear problems has also been closely related to the development of computer science.In mathematics and physics,people are accustomed to using equations to describe the laws of motion of objective things.Nonlinear partial differential equation is a mathematical physical equation which can describe the nonlinear evolution process of objective things.However,there is almost no general solutions to solve such equations in mathematics.In this paper,we use an auxiliary function method to solve the exact traveling wave solutions for the nonlinear Benjamin-Bona-Mahonye equation.With the help of the symbolic computation system Maple,together with adjusting the value of parameter m in the auxiliary function method,then abundant travelling wave solutions of the BBM equation are obtained.We compare the numbers and the forms of the solutions under two cases.It shows that the modified auxiliary function method can help us to obtain more traveling wave solutions.