Fourier–Taylor parameterization of unstable manifolds for parabolic partial differential equations: Formalism, implementation and rigorous validation

Christian Reinhardt, J. D. Mireles James*

*Corresponding author for this work

Research output: Contribution to JournalArticleAcademicpeer-review

Abstract

We study polynomial expansions of local unstable manifolds attached to equilibrium solutions of parabolic partial differential equations. Due to the smoothing properties of parabolic equations, these manifolds are finite dimensional. Our approach is based on an infinitesimal invariance equation and recovers the dynamics on the manifold in addition to its embedding. The invariance equation is solved, to any desired order in space and time, using a Newton scheme on the space of formal Fourier–Taylor series. Under mild non-resonance conditions we show that the formal series converge in some small enough neighborhood of the equilibrium. An a-posteriori computer assisted argument is given which, when successful, provides mathematically rigorous convergence proofs in explicit and much larger neighborhoods. We give example computations and applications.

Original languageEnglish
Pages (from-to)39-80
Number of pages42
JournalIndagationes Mathematicae
Volume30
Issue number1
Early online date23 Aug 2018
DOIs
Publication statusPublished - Jan 2019

Keywords

  • Computer-assisted proof
  • Connecting orbit
  • Parabolic partial differential equations
  • Parameterization method
  • Unstable manifold

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