Justification of the Nonlinear Schr\"odinger approximation for a quasilinear Klein-Gordon equation

TL;DR

This paper proves the validity of the nonlinear Schrödinger approximation for a quasilinear Klein-Gordon equation by establishing error estimates in Sobolev norms, marking the first such proof for a nonlinear hyperbolic system.

Contribution

It introduces a novel method to rigorously justify the NLS approximation for quasilinear hyperbolic equations with quadratic terms, extending previous results.

Findings

First proof of NLS approximation validity with error estimates in Sobolev spaces.

Method applicable to other quasilinear hyperbolic systems.

Establishes rigorous bounds between exact and approximate solutions.

Abstract

We consider a nonlinear Klein-Gordon equation with a quasilinear quadratic term. The Nonlinear Schr\"odinger (NLS) equation can be derived as a formal approximation equation describing the evolution of the envelopes of slowly modulated spatially and temporarily oscillating wave packet-like solutions to the quasilinear Klein-Gordon equation. It is the purpose of this paper to present a method which allows one to prove error estimates in Sobolev norms between exact solutions of the quasilinear Klein-Gordon equation and the formal approximation obtained via the NLS equation. The paper contains the first validity proof of the NLS approximation of a nonlinear hyperbolic equation with a quasilinear quadratic term by error estimates in Sobolev spaces. We expect that the method developed in the present paper will allow an answer to the relevant question of the validity of the NLS approximation for other quasilinear hyperbolic systems.