一类带有慢变参数的sine-Gordon方程的单脉冲异宿轨道

One-Pulse Travelling Front Solutions of a sine-Gordon Equation with Slowly Varying Parameters

基于Fenichel的几何奇异摄动理论,结合Melnikov方法,该文研究一类带慢变参数的sine-Gordon方程单脉冲波前解的存在性.首先,基于几何奇异摄动理论进行快慢分离,获得层系统和退化系统及其动力学;接着,引入Melnikov函数度量慢流形的稳定和不稳定流形的横截相交性,获得Take-off和Touch-down曲线的解析式.控制Take-off和Touch-down曲线使之分别与两个慢流形上鞍点的不稳定和稳定流形横截相交,从而得到奇异异宿轨道的存在性.经摄动,在该奇异异宿轨附近可获得异宿于系统两个不同鞍点的异宿轨道的存在性,从而上述带慢变参数的sine-Gordon方程的单脉冲波前解的存在性可得.最后,考虑了一个具体的例子,验证理论结果的正确性.

By combining Fenichel's geometric singular perturbation theory and Melnikov func-tion method,this paper studies the existence of l-pulse travelling front solutions of a sine-Gordon equation with slowly varying.parameters.Firstly,we get the layer system and thereduced system respectively as well as their global dynamics via the technique of fast-slowseparation,and then,we introduce the.Melnikov function to determine the transversal inter-sections between the stable and unstable manifolds of the slow manifold,where we define theso-called Take-off and Touch-down curves.By controlling the Take-off and Touch-down curvesto respectively intersect with the stable and unstable manifolds of the saddle points on theslow manifolds transversally,we get the singular heteroclinic orbits with transversality.Cor-respondingly we get the existence of heteroclinic orbits of the full singularly perturbed systemby perturbing such singular heteroclinic orbits.Finally,we consider an example to verify the correctness of the obtained the oretical results.