Dimension Breaking from Spatially-Periodic Patterns to KdV Planforms

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• 作者：Thomas J. Bridges
• 关键词：Lagrangian ; Bifurcation ; Patterns ; Multi ; pulse ; Modulation ; Elliptic PDEs
• 刊名：Journal of Dynamics and Differential Equations
• 出版年：2015
• 出版时间：December 2015
• 年：2015
• 卷：27
• 期：3-4
• 页码：443-456
• 全文大小：501 KB
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• 作者单位：Thomas J. Bridges (1)
  
  1. Department of Mathematics, University of Surrey, Guildford, GU2 7XH, England, UK
  
• 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

文摘

The problem of dimension breaking, for gradient elliptic partial differential equations in the plane, from a family of one-dimensional spatially periodic patterns (rolls) is considered. Conditions on the family of rolls are determined that lead to dimension breaking in the plane governed by a KdV equation relative to the periodic state. Since the KdV equation is time-independent, the \(N\)-pulse solutions of KdV provide a sequence of multi-pulse planforms in the plane bifurcating from the rolls. The principal examples are the nonlinear Schr?dinger equation, with evolution in the plane, and the steady Swift–Hohenberg equation with weak transverse variation. Keywords Lagrangian Bifurcation Patterns Multi-pulse Modulation Elliptic PDEs