基于近似广义对称的扰动非线性方程的若干问题

Nonlinear partial differential equations characterize many nonlinear phenomena which arise from various fieldes including natural science field and social science field and so on. With the development of nonlinear science, study on symmetries，solutions and other problems of the perturbed nonlinear partial differential equations has been attracting more and more attentions of many researchers. In this project, we put stress on the study of some issues concerning the approximate generalized symmetries, the approximate solutions and the approximate conservation laws of the perturbed nonlinear equations. The project contains the following main aspects: (1) Develop the derivative-dependent functional varible separation approach to the perturbed nonlinear equations. (2) Propose the general theory and methods to reduce and solve the perturbed nonlinear equations in terms of the approximate generalized conditional symmetries. (3) Establish the fundamental theory and methods, which can reduce perturbed nonlinear equations correspondingly to the problems of initial-value, initial boundary-value and free boundary-value's of some low dimensional perturbed nonlinear equations. (4) Develop and complete the theory and methods to derive the approximate generalized symmetries and the approximate conservation laws for the perturbed nonlinear equations. The results of the study on these leading subjects will certainly enrich the classical theories of mathematics, physics, etc. and present a series of new approaches of nonlinear mathematical physics for us to solve some theoretical problems or practical ones well.

非线性偏微分方程刻画了自然科学和社会科学等领域的非线性现象。随着非线性科学的不断发展，扰动非线性偏微分方程的对称、求解及其相关问题，已成为众多科技工作者研究的热点。本项目着重研究扰动非线性方程的近似广义对称、近似解与近似守恒律。主要包括：（1）建立和完善扰动非线性方程的导数相关泛函分离变量法；（2）建立和完善扰动非线性方程的近似广义条件对称和近似求解的一般理论和方法；（3）建立和完善扰动非线性方程可近似广义条件对称约化为低维扰动方程的初值、初边值和自由边值问题的基本理论和方法；（4）发展和完善扰动非线性方程的近似广义对称与近似守恒律的理论和方法。这些前沿课题的研究成果将会丰富数学、物理学等的经典理论并有望建立新的、行之有效的数学物理方法，以便提供更好地解决实际问题的理论和手段。

非线性偏微分方程刻画了自然科学和社会科学等领域的非线性现象。随着非线性科学的不断发展，非线性偏微分方程的对称、求解及其相关问题，已成为众多科技工作者研究的热点。本项目着重研究了若干非线性方程的不变集、不变子空间、对称分类及求解、守恒律等问题，并将一些理论推广到时间分数阶演化系统。此外，还对一些非线性生物模型和化学模型进行了定性研究。取得了以下主要成果：（1）证明了一类(1 + 2)-维波动方程在特定不变集下存在某种特解。给出了三阶平方算子容许极大维不变子空间的完全分类，进而导出了具有该算子的演化方程的一些特解；（2）给出了二阶非线性演化方程在相关变换的半单群下的完全分类，给出了KdV型方程在Galilei对称下不变方程及不变解解的完全分类。更进一步，将群理论和相容Riccati展开法推广应用到几类时间分数阶演化系统；（3）获得了(2+1)维非线性非色散长波 (DLW) 系统的非局部对称、相互作用解和无穷多守恒律。证明了(2+1)维修正的色散水波(MDWW)系统的非线性自伴随性，获得了其守恒律和Soliton-Cnoidal波相互作用解；（4）分别对Lotka-Volterra捕食-食饵模型、Lengyel-Epstein反应扩散模型进行了定性分析研究。这些前沿课题的研究成果将会丰富数学、物理学、生物学等领域的经典理论并通过创建这些新的、有效的数学物理方法，以便提供更好地解决实际问题的理论和手段。