一类反应扩散方程的孤立周期波和局部临界周期分支

Solitary Periodic Waves and Local Bifurcations of Critical Periods for a Class of Reaction-Diffusion Equations

研究了一类含有五次非线性反应项和常数扩散项的反应扩散方程的小振幅孤立周期波解,以及它的行波方程局部临界周期分支问题.运用行波变换将反应扩散方程转换为对应的行波系统,应用奇点量方法和计算机代数软件MATHEMATICA计算出该系统的前8个奇点量,得到该系统奇点的两个中心条件,并证明行波系统原点处可分支出8个极限环,对应的非线性反应扩散方程存在8个小振幅孤立周期波解;通过周期常数的计算,得到了行波系统原点的细中心阶数,并证明该系统最多有3个局部临界周期分支,且能达到3个局部临界周期分支;通过分析行波系统的临界周期分支,得到该反应扩散方程有3个临界周期波长.

The small-amplitude solitary periodic wave solutions and the local critical periodic bifurcations of the traveling wave equations for a class of reaction-diffusion equations with quintic nonlinear reaction terms and constant diffusion terms were studied.First,the reaction-diffusion equation was transformed into the corresponding traveling wave system through traveling wave transformation.The first 8 singular point quantities of the system were calculated with the singular point value method and the computer algebra software MATHEMATICA.Then,2 center conditions for the singular point of the system were obtained,8 limit cycles were proved to bifurcate at the origin of the traveling wave system,and 8 small-amplitude solitary periodic wave solutions were found to exist in the corresponding nonlinear reaction-diffusion equation.Furthermore,through computation of the period constants,the weak center order for the origin of the traveling wave system was derived.Then,the system was proved to have at most 3 local critical periodic bifurcations and be able to reach the 3 bifurcations.Moreover,the analysis of the critical periodic bifurcations of the traveling wave system reveals that the reaction-diffusion equation has 3 critical periodic wavelengths.