非线性数学物理中可积Ermakov系统的研究

项目名称： 非线性数学物理中可积Ermakov系统的研究

项目编号： No.11301269

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 安红利

作者单位： 南京农业大学

项目金额： 22万元

中文摘要： 发展非线性方程精确求解的理论和方法是数学物理的核心问题之一。近年来，研究发现，通过构造非线性数学物理模型中的可积Ermakov系统的办法可以达到求解一些非线性方程的目的。Navier-Stokes方程、非线性Schrodinger方程和磁流体方程是非线性科学中应用性和普适性很强的三个基本方程,广泛应用于诸多重要的物理分支。本项目将借助Ermakov系统来研究与上述三个方程密切相关的重要数学物理方程的精确解和动力学行为：首先，根据模型的特点，建立不同的构建Ermakov系统的理论；其次，寻求约化方程潜在的不变量，再结合Ermakov系统所固有的不变量来构造模型的精确解；最后，对所得精确解进行数值模拟和图形分析，研究它们的稳定性、周期性等动力学行为，并揭示解所对应的物理现象。本项目的实施，将为一些重要非线性数学物理方程的求解及其在实际物理中的应用提供一些可能的指导或参考。

中文关键词： 非线性方程；可积系统；可积Ermakov系统；纳维叶-斯托克斯方程；非线性薛定谔方程

英文摘要： It is one of the key problems of mathematical physics to develop theories and methods of exact solutions for nonlinear equations. Recently, it is shown that one can obtain the solutions of some nonlinear equations by constructing the integrable Ermakov systems underlying them. The Navier-Stokes equation、nonlinear Schrodinger equation and magnetohydrodynamic equation are three basic equations of nonlinear science, which have strong applications and universalities. They have been widely used in many important branches of physics. In this project, the Ermakov systems will be used to investigate the exact solutions and dynamic behaviors of some important mathematical physics equations, which are closely related to the above three equations. Firstly,different theories on constructing Ermakov systems will be developed according to the features of equations. Secondly, exact solutions will be constructed by using the invariants that come from the reduced equations and Ermakov systems itself. Finally, numerical simulations and figures will be used to study the dynamic behaviors (such as stability、periodicity) and to illustrate the corresponding physical phenomena. The implementation of this project may provide some guidance or reference for seeking solutions of some important nonlinear mathematical physics equations and

英文关键词： Nonlinear equations；Integrable system；Integrable Ermakov system；Navier-Stokes equation；Nonlinear Shrodinger equation