부산대학교 도서관

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 Schr\"odinger (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.
Comment: arXiv admin note: text overlap with arXiv:1602.08016. This paper is a minor revision of the previous submission arXiv:1605.08704v1. In its present form, the paper has been accepted for publication in Journal of Differential Equations