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• W2405744903 abstract "We consider a nonlinear dispersive equation with a quasilinear quadratic term. We establish two results. First, we show that solutions to this equation with initial data of order $mathcal{O}(varepsilon)$ in Sobolev norms exist for a time span of order $mathcal{O}(varepsilon^{-2})$ for sufficiently small $varepsilon$. Secondly, we derive the Nonlinear Schrodinger (NLS) equation as a formal approximation equation describing slow spatial and temporal modulations of the envelope of an underlying carrier wave, and justify this approximation with the help of error estimates in Sobolev norms between exact solutions of the quasilinear equation and the formal approximation obtained via the NLS equation. The proofs of both results rely on estimates of appropriate energies whose constructions are inspired by the method of normal-form transforms. To justify the NLS approximation, we have to overcome additional difficulties caused by the occurrence of resonances. We expect that the method developed in the present paper will also allow to prove the validity of the NLS approximation for a larger class of quasilinear dispersive systems with resonances." @default.
• W2405744903 created "2016-06-24" @default.
• W2405744903 creator A5060796178 @default.
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• W2405744903 date "2018-02-01" @default.
• W2405744903 modified "2023-10-17" @default.
• W2405744903 title "Existence of long time solutions and validity of the nonlinear Schrödinger approximation for a quasilinear dispersive equation" @default.
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• W2405744903 doi "https://doi.org/10.1016/j.jde.2017.10.031" @default.
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