非线性系统可积性的若干机械化算法及应用研究

项目名称： 非线性系统可积性的若干机械化算法及应用研究

项目编号： No.11201290

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 徐桂琼

作者单位： 上海大学

项目金额： 22万元

中文摘要： 将数学机械化的原理和思想引入到非线性系统可积性和精确解研究中，与孤子理论中的Painlevé法和Hirota方法结合起来，建立和发展可积性判定、可积性质推导及精确求解的若干机械化算法。以符号计算为工具，编制可积性与精确求解的自动推导软件包。本项目的研究成果将拓宽数学机械化在微分领域的应用范围，为相关学科的研究提供实用方便的研究工具。研究内容包括：（1）发展非等谱发展方程Painlevé的构造性算法，并推广到非线性差分-微分方程。（2）建立双线性形式和双线性B?cklund变换、Lax对和守恒律的自动推导算法。（3）研究连续系统和离散系统的拟周期解、ripplon解、dromion解等的机械化算法。（4）基于Maple编制软件包，实现非等谱发展方程和差分-微分方程Painlevé检验、可积性质推导和精确求解等功能。

中文关键词： 数学机械化；非线性系统；可积性；精确解；

英文摘要： In this project, the theory of mathematics mechanization proposed by famous mathematician Wu Wentsun is widely used to the constructive study of integrability and exact solutions for nonlinear systems. Based on the Painlevénalysis and Hirota's bilinear method resulting from the soliton theory, a number of mechanical algorithms will be proposed for testing and deriving integrable properties. At the same time, Ritt-Wu's method is used correctly in some key steps while designing the algorithms. With the aid of symbolic computation, several packages will be developed for deriving integrabilities and exact solution automatically. The algorithms and the relevant packages presented in this project provide a very effective tool for the study of the problem of nonlinear physics and mathematics resulting from the modern science and technology. The subjects of this project mainly include: (1) present a constructive algorithm of the Painlevéest and Painlevélassification for nonlinear evolution equations with arbitrary coefficient functions, in addition, the which will be extended to nonlinear difference-differential equations. (2)propose a systematic algorithm for constructing bilinear forms, bilinear Backlund transformations, Lax pairs and conservation laws. (3) investigate the mechanical algorithms for deriving abun

英文关键词： mathematics mechanization；nonlinear system；integrability；exact solutions；