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

2022/06/23 by Roberto Feola, Feola, Roberto, Filippo Giuliani +1
Mathematics · Physics and Astronomy · #35L725 #35Q5 #37K06 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Photonic Systems #Numerical methods for differential equations

paper · pdf · doi:10.48550/arxiv.2206.11836

openalex publication_date 2022/06/23 · openalex created_date 2022/06/26 · openalex updated_date 2026/07/28

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:=ℝ/(2πL ℤ), 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.

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