时间分数阶物理方程的行波解研究

A model of time-fractional partial differential equations is one of important mathematical models to describe the complex physical and Mechanical phenomena. Obtaining traveling wave solutions and investigating qualitative behaviors for the model would help find out the dynamical properties and the law of physical and mechanical process. This project targets several types of time-fractional order nonlinear equations originating from Physics and Mechanics, applies the combined methods of Lie Group transformation and dynamical system theories, and studies the invariant solutions , traveling wave solutions and the evolution of dynamical behaviors for solutions. For given fractional equations, such as Zakharov-Kuznetsov equation, Fokker-Planke quation, Konopelchenko-Dubrovsky equation, Benjamin-Ono equation, etc., this project focuses on two aspects. The first is that Lie Group method in solving integral partial differential equations is extended to the case of fractional partial differential equations. Moreover, the homogeneous principle hidden in the process of finding solutions is generalized. The second is that dynamical system method is applied to find some traveling solutions of fractional partial differential equations. The dynamical system method, combined with Lie Group transformation method is used to explore the presentation of corresponding traveling wave solutions in the case of fractional partial differential equations, and the evolutionary process of dynamical behaviors for specific traveling wave solutions with the order of a derivative from an integer to a fraction. Furthermore, a new idea is obtained, in which the dynamical system method is used to research the traveling wave solutions of fractional partial differential equations.

时间分数阶偏微分方程模型是描述复杂物理、力学问题的重要数学模型之一。其行波解的求解和定性行为的研究，将有助于揭示复杂力学与物理过程的动力学性质及其规律。本项目以几类源于物理、力学中的分数阶非线性方程为对象，用李群变换与动力系统理论相结合的方法，研究几类非线性方程的不变解、行波解求解及解的动力学行为演化问题。针对给定的分数阶方程，如Zakharov-Kuznetsov、Fokker-Plank、Konopelchenko-Dubrovsky、Benjamin-Ono方程等，本项目着力于两个方面的研究。一是将求解整数阶偏微分方程的李群方法推广到分数阶偏微分方程的情形，并将蕴含其中的齐次原理进行推广；二是用动力系统方法求解整数阶方程的行波解，并与李群变换法相结合，来探索分数阶方程相应行波解的表示及其动力学行为随求导阶数变化的规律，为用动力系统方法研究分数阶偏微分方程的行波解问题提供一种新思路。

为了把研究整数阶非线性偏微分方程行波解的李群变换法（代数方法）和动力系统方法（几何方法）推广到时间分数阶方程情形，项目组综合用李群变换法、动力系统方法、贝克隆变换法等方法研究了5类整数阶非线性偏微分方程的不变解。一方面，综合应用李群变换法和动力系统方法，分别得到了一类生物趋化模型的有生物意义的三类解及其分支行为；一类七阶KdV方程的三个守恒律；广义Burgers方程的李对称分析、扭波解的动力学行为；另一方面，综合应用李群变换法、贝克隆变换法、二元贝尔多项式法及微分限制技巧，得到了Hirota双线性方程块解的显式表示及其间演变的动力学行为；Bogoyavlenskii-Kadomtsev-Petviashvili 方程的复合波解、块波解的显式表示与动力学行为。这些结果丰富了非线性偏微分方程求解的方法和理论，将为理解复杂的物理现象提供科学参考。. 通过本项目的支持，项目组在国内外重要核心期刊发表第一标注论文5篇，其中4篇被SCI收录，培养了2名硕士研究生。