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Multi-Pulse Evolution and Space-Time Chaos in Dissipative Systems
Sergey Zelik, University of Surrey, Guildford, United Kingdom, and Alexander Mielke, Weierstrass Institut für Angewandte Analysis und Stochastik, Berlin, Germany
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Memoirs of the American Mathematical Society
2009; 97 pp; softcover
Volume: 198
ISBN-10: 0-8218-4264-1
ISBN-13: 978-0-8218-4264-5
List Price: US$65
Individual Members: US$39
Institutional Members: US$52
Order Code: MEMO/198/925
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The authors study semilinear parabolic systems on the full space \({\mathbb R}^n\) that admit a family of exponentially decaying pulse-like steady states obtained via translations. The multi-pulse solutions under consideration look like the sum of infinitely many such pulses which are well separated. They prove a global center-manifold reduction theorem for the temporal evolution of such multi-pulse solutions and show that the dynamics of these solutions can be described by an infinite system of ODEs for the positions of the pulses.

As an application of the developed theory, The authors verify the existence of Sinai-Bunimovich space-time chaos in 1D space-time periodically forced Swift-Hohenberg equation.

Table of Contents

  • Introduction
  • Assumptions and preliminaries
  • Weighted Sobolev spaces and regularity of solutions
  • The multi-pulse manifold: general structure
  • The multi-pulse manifold: projectors and tangent spaces
  • The multi-pulse manifold: differential equations and the cut off procedure
  • Slow evolution of multi-pulse profiles: linear case
  • Slow evolution of multi-pulse structures: center manifold reduction
  • Hyperbolicity and stability
  • Multi-pulse evolution equations: asymptotic expansions
  • An application: spatio-temporal chaos in periodically perturbed Swift-Hohenberg equation
  • Bibliography
  • Nomenclature
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