Renormalization group and the Melnikov problem for PDE's

Bricmont, Jean;Kupiainen, Antti;Schenkel, A
(2001) Communications in Mathematical Physics — Vol. 221, n° 1, p. 101-140 (2001)

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Authors
  • Bricmont, JeanUCLouvain
    Author
  • Kupiainen, AnttiUCLouvain
    Author
  • Schenkel, A
    Author
Abstract
We give anew 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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Bricmont, J., Kupiainen, A., & Schenkel, A. (2001). Renormalization group and the Melnikov problem for PDE’s. Communications in Mathematical Physics, 221(1), 101-140. https://doi.org/10.1007/s002200100471 (Original work published 2001)