Solitary waves generated by the interaction of oscillating nonlinear waves
type: Item
instance of: scholarly article
title: Solitary waves generated by the interaction of oscillating nonlinear waves
zbMATH Open document ID: 0909.35087
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 1996-07-16T00:00:00Z
review text: The paper deals with rigorous weakly nonlinear geometrical optics. It is devoted to the precise description of the solutions of 1D nonlinear hyperbolic systems with high frequency oscillating initial data and brings to the fore a new phenomenon: the creation of solitary waves by nonlinear interaction of oscillating waves. It is a resonance phenomenon called ``weak resonance'' which corresponds to a stationary property of a linear combination of the phases brought in play by the initial data oscillations. Such weak resonances can be the cause of singular behaviors of the solution like cahotic or explosive behaviors. Several detailed examples are given. This work is based on a rigorous analysis of the fluctuations of the solution and their nonlinear evolution, and uses techniques introduced by \textit{J. L. Joly}, \textit{G. Métivier} and \textit{J. Rauch} [J. Funct. Anal. 114, No. 1, 106-231 (1993; Zbl 0851.35023)]. The results concern local in time classical solutions and apply before the shock waves formation in the quasilinear case and the \(L^\infty\) explosion in the semilinear one.
Mathematics Subject Classification ID: 35L60
Mathematics Subject Classification ID: 35C20
zbMATH DE Number: 891040
zbMATH Keywords: weak resonance
zbMATH Keywords: weakly nonlinear geometrical optics
zbMATH Keywords: oscillating initial data
zbMATH Keywords: creation of solitary waves
author: Olivier Guès
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 891040