First and second order approximations for a nonlinear wave equation

2011/11/25 by Oana Pocovnicu, Pocovnicu, Oana · 1 citation
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #math.AP

paper · pdf · doi:10.48550/arxiv.1111.6060

30 pages

arxiv created 2011/11/25 · openalex publication_date 2011/11/25 · arxiv updated 2011/11/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the nonlinear wave equation (NLW) iv_t-|D|v=|v|²v on the real line. We show that in a certain regime, the resonant dynamics is given by a completely integrable nonlinear equation called the Szego equation. Moreover, the Szego equation provides a first order approximation for NLW for a large time. The proof is based on the renormalization group method of Chen, Goldenfeld, and Oono. As a corollary, we give an example of solution of NLW whose high Sobolev norms exhibit relative growth. An analogous result of approximation was proved by Gerard and Grellier on the torus using the theory of Birkhoff normal forms. We improve this result by finding the second order approximation on the torus with the help of an averaging method introduced by Temam and Wirosoetisno. We show that the effective dynamics will no longer be given by the Szego equation.

Citations

• Global existence for coupled Klein-Gordon equations with different speeds
• The Renormalization Group and Singular Perturbations: Multiple-Scales, Boundary Layers and Reductive Perturbation Theory

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• Long-time instability and unbounded Sobolev orbits for some periodic nonlinear Schrödinger equations

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