数学物理学报 ›› 2019, Vol. 39 ›› Issue (6): 1365-1375.

• 论文 • 上一篇    下一篇

一类广义Gierer-Meinhardt方程多脉冲同宿解的再研究

朱昆1,沈建和1,2,*()   

  1. 1 福建师范大学数学与信息学院 福州 350117
    2 福建省分析数学及其应用重点实验室 福州 350117
  • 收稿日期:2018-05-03 出版日期:2019-12-26 发布日期:2019-12-28
  • 通讯作者: 沈建和 E-mail:jhshen@fjnu.edu.cn
  • 基金资助:
    国家自然科学基金(11771082);福建省教育厅新世纪杰青项目

A Revisit on Multiple-Pulse Homoclinic Solutions in a Generalized Gierer-Meinhardt Equation

Kun Zhu1,Jianhe Shen1,2,*()   

  1. 1 College of Mathematics and Informatics, Fujian Normal University, Fuzhou 350117
    2 Fujian Key Laboratory of Mathematical Analysis and Applications, Fujian Normal University, Fuzhou 350117
  • Received:2018-05-03 Online:2019-12-26 Published:2019-12-28
  • Contact: Jianhe Shen E-mail:jhshen@fjnu.edu.cn
  • Supported by:
    the NSFC(11771082);the Program for New Century Excellent Talents in Fujian Province University

摘要:

关于广义Gierer-Meinhardt(G-M)方程多脉冲同宿轨道,Doelman等[Indiana Univ Math J,2001,50:443-507]已进行了详细的研究,获得了存在性和稳定性及其参数条件.然而,在上述Doelman等的工作中,Melnikov积分(度量层系统的临界流形的稳定和不稳定流形的横截相交性)并没有计算.因此,该文的工作有两个方面:首先,通过初等积分法,计算获得一类与层系统相关的二阶非线性保守方程同宿轨道的显式表达式;接着,基于该显式表达式,对Melnikov积分进行详细的计算,从而获得上述广义G-M方程存在多脉冲同宿轨道的更为精细的参数条件.

关键词: 广义G-M方程, 多脉冲同宿轨道, Melnikov函数

Abstract:

In paper[1] (Indiana Univ Math J, 2001, 50:443-507), the authors studied the existence and stability of multiple-pulse homoclinic solutions in a generalized Gierer-Meinhardt equation. However, in this paper, a general integral measuring the distance of the stable and unstable manifolds of the critical manifold of the layer system, i.e., the Melnikov integral, was not computed explicitly. So we have two aims in this manuscript. Firstly, we give an elementary method to solve a second-order nonlinear conservative system and hence obtain the explicit representation of the homoclinic orbit. Secondly, we substitute the explicit representation of the homoclinic orbit into the the Melnikov integral. By computing such a general integral, we obtain a more explicitly parametric condition on the existence of multiple-pulse homoclinic solutions in such a generalized Gierer-Meinhardt equation.

Key words: Generalized Gierer-Meinhardt equation, Multi-pulse homoclinic orbit, Melnikov function

中图分类号: 

  • O175.12