Existence of long time solutions and validity of the Nonlinear Schr\"odinger approximation for a quasilinear dispersive equation

TL;DR

This paper proves the existence of long-time solutions for a quasilinear dispersive equation and rigorously justifies the nonlinear Schrödinger approximation for slow modulations, using energy estimates and normal-form transforms.

Contribution

It establishes long-time solutions and provides a rigorous justification of the NLS approximation for a class of quasilinear dispersive equations with resonances.

Findings

Solutions exist for time scales of order (\u03b5^{-2})

NLS provides a valid approximation for slow modulations

Error estimates confirm the approximation's accuracy

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 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.