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Renormalization Group and the Melnikov Problem for PDE's

Jean Bricmont·Antti Kupiainen·Alain Schenkel
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Abstract

We give a new proof of persistence of quasi-periodic, low dimensional elliptic tori in infinite dimensional systems. The proof is based on a renormalization group iteration that was developed recently in [BGK] to address the standard KAM problem, namely, persistence of invariant tori of maximal dimension in finite dimensional, near integrable systems. Our result covers situations in which the so called normal frequencies are multiple. In particular, it provides a new proof of the existence of small-amplitude, quasi-periodic solutions of nonlinear wave equations with periodic boundary conditions.

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Jean Bricmont, Antti Kupiainen, Alain Schenkel. 2001-02-28. Renormalization Group and the Melnikov Problem for PDE's. https://doi.org/10.1007/s002200100471

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