Cahn-Hilliard-Schlgl反应-扩散方程分叉问题

BIFURCATIONS OF CAHN-HILLIARD-SCHLOGL REACTION-DIFFUSION EQUATION

本文利用Lyapunov-Schmidt约化方法、奇异性理论及摄动方法,对服从Cahn-Hilliard广义扩散定律的带有Schlgl反应项的Cahn-Hilliard-Schlgl反应-扩散方程的分叉情况进行了研究。研究结果表明,C-H-S方程可以产生单模态的空间周期一次和二次分叉。特别地,经小扰动的退化的C-H-S方程,在不同条件下,还可能产生不同形式的更为复杂的双模态混合的二次分叉。本文还给出了分叉类型和一次分叉解的近似表达式。

Bifurcation of Cahn-Hilliard-Schlogl reaction diffusion equations,in which the reaction term is Schlogl's one and the diffusion term is governed by Cahn-Hilliard's generalized diffusion law,are studied by means of Lyapunov-Schmidt reduction method.singularity theory and perturbation method.It is shown that C-H-S equation have spatially periodic primary and secondary bifurcations with simple mode.Especially,the perturbed degenerate D-H-S equation,under different conditions,may have different kinds of double mixed-mode.The kinds of the bifurcations and ap-poraximate expressions of this primary bifurcations are given in the paper.