First and Second Order Approximations for a Nonlinear Wave Equation

Abstract

In this paper we consider the following nonlinear half-wave equation: NLW $$\begin{aligned} i\partial _t\mathcal{V }-|D|\mathcal{V }=|\mathcal{V }|^2\mathcal{V }, \end{aligned}$$ where $$D = -i \partial _{x}$$ , both on $$\mathbb{R }$$ and $$\mathbb{T }$$ . On $$\mathbb{R }$$ , we prove that, if the initial condition is of order $$O({\varepsilon })$$ and supported on positive frequencies only, then the corresponding solution can be approximated by the solution of the Szegő equation. The Szegő equation $$i\partial _tu=\Pi _+(|u|^2u)$$ , where $$\Pi _+$$ is the Szegő projector onto non-negative frequencies, is a completely integrable system that gives an accurate description of solutions of (NLW). The approximation holds for a long time $$0\le t\le C{\varepsilon }^{-2}\big [\log (1/{\varepsilon }^{\delta })\big ]^{1-2\alpha }$$ , $$0\le \alpha \le 1/2$$ . The proof is based on the renormalization group method. As a corollary, we give an example of a solution of (NLW) on $$\mathbb{R }$$ whose high Sobolev norms grow over time, relative to the norm of the initial condition. An analogous result of approximation was proved by Gerard and Grellier (Anal PDEs, arXiv:1110.5719v1) on $$\mathbb{T }$$ using Birkhoff normal forms. We improve their result by finding a second order approximation with the help of an averaging method. We show, in particular, that the effective dynamics is no longer given by the Szegő equation.