Long time NLS approximation for the quasilinear Klein-Gordon equation on large domains under periodic boundary conditions

TL;DR

This paper rigorously justifies the nonlinear Schrödinger (NLS) approximation for quasilinear Klein-Gordon equations on large periodic domains, over long times, by addressing resonances and small divisors with advanced analytical techniques.

Contribution

It extends the validity of the NLS approximation to long times for quasilinear Klein-Gordon equations on large domains, using second-order analysis and sophisticated mathematical tools.

Findings

NLS approximation holds beyond cubic nonlinear time scale

Successful handling of quasi-resonances and small divisors

Application of para-differential calculus and normal form methods

Abstract

We provide the rigorous justification of the NLS approximation, in Sobolev regularity, for a class of quasilinear Hamiltonian Klein Gordon equations with quadratic nonlinearities on large one-dimensional tori $\T_L:=\mathbb{R}/(2\pi L \mathbb{Z})$, $L\gg 1$. We prove the validity of this approximation over a \emph{long-time} scale, meaning that it holds beyond the cubic nonlinear time scale. To achieve this result we need to perform a second-order analysis and deal with higher order resonant wave-interactions. The main difficulties are provided by the quasi-linear nature of the problem and the presence of small divisors arising from quasi-resonances. The proof is based on para-differential calculus, energy methods, normal form procedures and a high-low frequencies analysis.