Coherent nonlinear waves and the Wiener algebra
type: Item
instance of: scholarly article
title: Coherent nonlinear waves and the Wiener algebra
zbMATH Open document ID: 0791.35019
publication date: 1994-07-18T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/75053
review text: We study oscillatory solutions of semilinear first order symmetric hyperbolic system \(Lu= f(t,x,u, \overline{u})\), with real analytic \(f\). The main advance in this paper is that it treats multidimensional problems with profiles that are almost periodic in \(T\), \(X\) with only the natural hypothesis of coherence. In the special case where \(L\) has constant coefficients and the phases are linear, the solutions have asymptotic description \[ u^ \varepsilon= U(t,x,t/\varepsilon, x/\varepsilon)+ o(1) \] where the profile \(U(t,x,T,X)\) is almost periodic in \(T\), \(X\). The main novelty in the analysis is the space of profiles which have the form \[ U= \sum_{\tau,\omega\in \mathbb{R}^{1+d}} U_{\tau,\omega}(t,x) e^{i(\tau T+\omega X)}, \qquad \sum \| U_{\tau,\omega} (t,x)\|_{C([0, \underline{t}]: H^ s(\mathbb{R}^ d))} <\infty. \] Thus, \(U\) is an element of the Wiener algebra as a function of the fast variables. The profile \(U\) is uniquely determined from the initial data of \(u^ \varepsilon\) by profile equations of standard form. An application to conical refraction where the characteristics have variable multiplicity is presented.
Mathematics Subject Classification ID: 35C20
Mathematics Subject Classification ID: 35B30
Mathematics Subject Classification ID: 35L60
Mathematics Subject Classification ID: 35Q60
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35L45
zbMATH DE Number: 566968
zbMATH Keywords: interaction of high frequency solutions
zbMATH Keywords: geometric optics
zbMATH Keywords: crystal optics
zbMATH Keywords: oscillatory solutions
zbMATH Keywords: semilinear first order symmetric hyperbolic system
zbMATH Keywords: constant coefficients
zbMATH Keywords: Wiener algebra
zbMATH Keywords: application to conical refraction
author: Jean-Luc Joly
author: Guy Métivier
author: Jeffrey Rauch
MaRDI profile type: Publication
OpenAlex ID: W2320100317
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DOI: 10.5802/AIF.1393
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 566968