Justification of the Nonlinear Schrödinger Approximation for a Quasilinear Klein–Gordon Equation

Justification of the Nonlinear Schrödinger Approximation for a Quasilinear Klein–Gordon Equation We consider a nonlinear Klein–Gordon equation with a quasilinear quadratic term. The Nonlinear Schrödinger (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 that 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Communications in Mathematical Physics Springer Journals

Justification of the Nonlinear Schrödinger Approximation for a Quasilinear Klein–Gordon Equation

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Publisher
Springer Journals
Copyright
Copyright © 2017 by Springer-Verlag GmbH Germany
Subject
Physics; Theoretical, Mathematical and Computational Physics; Mathematical Physics; Quantum Physics; Complex Systems; Classical and Quantum Gravitation, Relativity Theory
ISSN
0010-3616
eISSN
1432-0916
D.O.I.
10.1007/s00220-017-2966-y
Publisher site
See Article on Publisher Site

Abstract

We consider a nonlinear Klein–Gordon equation with a quasilinear quadratic term. The Nonlinear Schrödinger (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 that 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.

Journal

Communications in Mathematical PhysicsSpringer Journals

Published: Jul 27, 2017

References

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