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

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 TL := R/(2πLZ), L ≫ 1. We prove the validity of this approximation over a 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. CONTENTS 2010 Mathematics Subject Classification. 35Q55 . Key words and phrases. Normal forms, Nonlinear Schrödinger equation, quasilinear Klein-Gordon, large tori. Acknowledgements. Roberto Feola and Filippo Giuliani have received funding from INdAM-GNAMPA Project CUP E55F 22000270001.