Effective integrable dynamics for some nonlinear wave equation

2011/10/26 by Patrick Gerard, Patrick Gérard, Gerard, Patrick +2
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Waves and Solitons #Quantum chaos and dynamical systems #math.AP

paper · pdf · doi:10.48550/arxiv.1110.5719

19 pages

arxiv created 2011/10/26 · openalex publication_date 2011/10/26 · arxiv updated 2011/10/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the following degenerate half wave equation on the one dimensional torus i∂_t u-|D|u=|u|²u, u(0,⋅)=u₀. We show that, on a large time interval, the solution may be approximated by the solution of a completely integrable system-- the cubic Szegö equation. As a consequence, we prove an instability result for large H^s norms of solutions of this wave equation.

Related

• First and second order approximations for a nonlinear wave equation
• The Szegö Cubic Equation
• Long time behavior of the solutions of NLW on the d-dimensional torus
• The Cubic Szeg\ho Equation with a Linear Perturbation
• Explicit formula for the solution of the Szegö equation on the real line and applications