2016/02/29 by Wolf-Patrick Düll · 2 citations
Mathematics · #math.AP
paper · pdf · doi:10.1007/s00220-017-2966-y
published as Comm. Math. Phys. 355 (2017), no. 3, 1189-1207 · This paper is a minor revision of the previous submission arXiv:1602.08016v1 with a slightly changed title. In its present form, the paper has been accepted for publication in Communications in Mathematical Physics
arxiv created 2017/06/06 · arxiv updated 2017/08/23
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 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.