基于Conley指标理论求解反应扩散方程的冲击波解  

Solving the shock wave solutions of reaction-diffusion equations based on Conley index theory

作　　者：张柳[1] 张晓丹[1] 

机构地区：[1]北京科技大学应用科学学院,北京100083

出　　处：《北京科技大学学报》2011年第2期250-256,共7页Journal of University of Science and Technology Beijing

基　　金：北京科技大学冶金工程研究院基础理论研究基金资助项目(No.00009503)

摘　　要：利用Conley指标理论研究一类非线性反应扩散方程的冲击波解的情况.以扩散系数作为反应扩散方程的参数,通过Conley指标和Morse分解分析行波解所满足的常微分方程的异宿轨道的存在性,并根据偏微分方程的孤立波与冲击波分别对应于常微分方程的同宿轨道与异宿轨道的思想,进而证明了反应扩散方程鞍-焦型、鞍-结型冲击波解的存在性.特别地,应用联络矩阵和传递矩阵可证明鞍-鞍型冲击波解的存在性和唯一性.使用Conley软件包和Maple软件编程计算了联络矩阵和传递矩阵.Based on Conley index theory, the shock wave solutions of a class of nonlinear reaction-diffusion equations were studied. Considering the diffusion coefficient as a system parameter, the existence of heteroclinic orbits of ordinary differential equations satisfied by traveling wave solutions is analyzed by using Conley index and Morse decompositions. The existence of saddle-focus and saddle-crunode style shock wave solutions of the reaction-diffusion equations is proved on the basis of an idea that the solitary waves and shock waves of partial differential equations correspond to the homoclinic orbits and heteroclinic orbits of ordinary differential equations. In particular, the existence and uniqueness of saddle-saddle style shock wave solutions are proved by using connection matrixes and transition matrixes, which are computed with Conley packages and Maple software by programming.

关 键 词：微分方程 反应扩散方程 冲击波 CONLEY指标 

分 类 号：O175.2[理学—数学]