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 language | English |
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Pages (from-to) | 39-80 |
Number of pages | 42 |
Journal | Indagationes Mathematicae |
Volume | 30 |
Issue number | 1 |
Early online date | 23 Aug 2018 |
DOIs | |
Publication status | Published - Jan 2019 |
Funding
We would like to thank Jonathan Jaquette for suggesting us the argument given in the Appendix . Conversations with J.B. van den Berg, J.P. Lessard and Rafael de la Llave were also extremely valuable. The final version of this manuscript was greatly improved thanks to the hard work and many invaluable comments of an anonymous referee, to whom we offer our sincere and heartfelt thanks. J.D.M.J. was partially supported by NSF grant DMS-1318172 . First author was partially supported by NWO . Appendix
Funders | Funder number |
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National Science Foundation | DMS-1318172 |
National Stroke Foundation | |
Nederlandse Organisatie voor Wetenschappelijk Onderzoek |
Keywords
- Computer-assisted proof
- Connecting orbit
- Parabolic partial differential equations
- Parameterization method
- Unstable manifold