Stability of Concatenated Traveling Waves

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• 作者：Xiao-Biao Lin ; Stephen Schecter
• 刊名：Journal of Dynamics and Differential Equations
• 出版年：2016
• 出版时间：September 2016
• 年：2016
• 卷：28
• 期：3-4
• 页码：867-896
• 全文大小：749 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Ordinary Differential Equations
  Partial Differential Equations
  Applications of Mathematics
  
• 出版者：Springer Netherlands
• ISSN：1572-9222
• 卷排序：28

文摘

We consider a reaction–diffusion equation in one space dimension whose initial condition is approximately a sequence of widely separated traveling waves with increasing velocity, each of which is individually asymptotically stable. We show that the sequence of traveling waves is itself asymptotically stable: as \(t\rightarrow \infty \), the solution approaches the concatenated wave pattern, with different shifts of each wave allowed. Essentially the same result was previously proved by Wright (J Dyn Differ Equ 21:315–328, 2009) and Selle (Decomposition and stability of multifronts and multipulses, 2009), who regarded the concatenated wave pattern as a sum of traveling waves. In contrast to their work, we regard the pattern as a sequence of traveling waves restricted to subintervals of \(\mathbb {R}\) and separated at any finite time by small jump discontinuities. Our proof uses spatial dynamics and Laplace transform.KeywordsInteraction of wavesReaction–diffusion equationSpatial dynamics Laplace transformExponential dichotomy in trace space