Effective integrable dynamics for a certain nonlinear wave equation

Abstract

We consider the following degenerate half-wave equation on the one-dimensional torus:We show that, on a large time interval, the solution may be approximated by the solution of a completely integrable system -the cubic Szegő equation. As a consequence, we prove an instability result for large H s norms of solutions of this wave equation.

Solvable Cubic Resonant Systems

“…An extreme case is the cubic Szegő equation [20], which corresponds to C nmkl = 1. This equation is Lax-integrable and has been thoroughly analysed [21][22][23] with very interesting results for its dynamics.…”

Benjamin–Ono soliton dynamics in a slowly varying potential

“…De Bouard and Debussche [9] considered a stochastic perturbation of KdV with multiplicative white noise. Pocovnicu [31,32] and Gérard & Grellier [12] considered the cubic-Szego equation. Mashkin [25,26,27] has considered perturbations of sine-Gordon kink solitons.…”

On the growth of Sobolev norms for the cubic Szegő equation

“…However, compared to the case α = 1, this equation is no more dispersive, which suggests that large time transition to high frequencies may be facilitated. In [11] -see also Pocovnicu [26]-, we proved that a Birkhoff normal form of this equation near the origin is given at first order by the following cubic Szegő equation,…”

“…In the limit case α = 1, Equation (1) is a nonlinear (half-) wave equation in one space dimension, which can be proved to be globally well-posed on H s for s ≥ 1 2 (see [11]). However, compared to the case α = 1, this equation is no more dispersive, which suggests that large time transition to high frequencies may be facilitated.…”