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arXiv · 2206.11836

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

Roberto Feola·Filippo Giuliani
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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.

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Roberto Feola, Filippo Giuliani. 2022-06-23. Long time NLS approximation for the quasilinear Klein-Gordon equation on large domains under periodic boundary conditions. https://arxiv.org/abs/2206.11836

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