非线性系统可积性的若干机械化算法及应用研究

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 Painlevé analysis 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 Painlevé test and Painlevé classification 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 abundant types of exact solutions such as quasi-periodic solution, ripplon solution, dromion solution, which are suitable for nonlinear partial differential equations and difference-differential equations. (4)with the aid of computer algebra system Maple, several software packages will be provided, which are used to perform the Painlevé test for nonisopectral evolution equations and difference-differential equations, and also derive several important integrabilities such as bilinear forms, bilinear Backlund transformations, Lax pairs and conservation laws automatically, but also deliver abundant types of exact solutions in a systematic way.

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分析法和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检验的构造性算法，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可积检验、可积性质推导和精确求解等功能。

本课题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分析法和Hirota 方法、Bell多项式方法、分离变量法和基方程展开法结合起来，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，获得了一些新的结果。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）提出并发展了广义非线性演化方程Painlevé可积归类的构造性算法，并将其推广到非等谱情形，研究了具有重要应用背景的广义破碎孤子方程、广义五阶KdV方程、广义七阶KdV方程，获得了若干新的可积模型；（2）对Painlevé可积的模型，充分利用截断展开法与Hirota双线性方法、Bell多项式和Nucci拟势方法相结合，探讨求得双线性形式、Bäcklund变换、Lax对和无穷守恒律的若干构造性算法。利用这些算法，对我们获得的新可积模型进行研究，得到了一系列新的可积特性；（3）发展Hirota双线性方法，构造非线性波方程的多孤子解、N波解、双周期解以及周期孤立波解等；（4）发展和提出了直接分离变量法，研究了（2+1）维mKdV方程、（3+1）维BKP方程、（3+1）破碎孤子方程等高维非线性方程，得到了其丰富的局域激发模式；（5）改进了基方程展开法，研究了具有强非线性项的几个非线性演化方程的精确解。