Coherent nonlinear waves and the Wiener algebra

Guy Métivier; Jean-Luc Joly; Jeffrey Rauch

Annales de l'institut Fourier (1994)

  • Volume: 44, Issue: 1, page 167-196
  • ISSN: 0373-0956

Abstract

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We study oscillatory solutions of semilinear first order symmetric hyperbolic system , with real analytic .The main advance in this paper is that it treats multidimensional problems with profiles that are almost periodic in with only the natural hypothesis of coherence.In the special case where has constant coefficients and the phases are linear, the solutions have asymptotic descriptionwhere the profile is almost periodic in .The main novelty in the analysis is the space of profiles which have the formThus, is an element of the Wiener algebra as a function of the fast variables.The profile is uniquely determined from the initial data of by profile equations of standard from.An application to conical refraction where the characteristics have variable multiplicity is presented.

How to cite

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Métivier, Guy, Joly, Jean-Luc, and Rauch, Jeffrey. "Coherent nonlinear waves and the Wiener algebra." Annales de l'institut Fourier 44.1 (1994): 167-196. <http://eudml.org/doc/75053>.

@article{Métivier1994,
abstract = {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\begin\{\} u^\varepsilon = U(t, x, t/\varepsilon , x/\varepsilon ) + o(1) \end\{\}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\begin\{\} U=\sum \_\{\tau , \omega \in \{\Bbb R\}^\{1+d\}\} U\_\{\tau , \omega \} (t, x)e^\{i(\tau T+\omega .X)\}, \sum \Vert U\_\{\tau , \omega \}(t, x)\Vert \_\{C([0, t]:H^s(\{\Bbb R\}^d))\}&lt; \infty .\end\{\}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 from.An application to conical refraction where the characteristics have variable multiplicity is presented.},
author = {Métivier, Guy, Joly, Jean-Luc, Rauch, Jeffrey},
journal = {Annales de l'institut Fourier},
keywords = {interaction of high frequency solutions; geometric optics; crystal optics; oscillatory solutions; semilinear first order symmetric hyperbolic system; constant coefficients; Wiener algebra; application to conical refraction},
language = {eng},
number = {1},
pages = {167-196},
publisher = {Association des Annales de l'Institut Fourier},
title = {Coherent nonlinear waves and the Wiener algebra},
url = {http://eudml.org/doc/75053},
volume = {44},
year = {1994},
}

TY - JOUR
AU - Métivier, Guy
AU - Joly, Jean-Luc
AU - Rauch, Jeffrey
TI - Coherent nonlinear waves and the Wiener algebra
JO - Annales de l'institut Fourier
PY - 1994
PB - Association des Annales de l'Institut Fourier
VL - 44
IS - 1
SP - 167
EP - 196
AB - 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\begin{} u^\varepsilon = U(t, x, t/\varepsilon , x/\varepsilon ) + o(1) \end{}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\begin{} U=\sum _{\tau , \omega \in {\Bbb R}^{1+d}} U_{\tau , \omega } (t, x)e^{i(\tau T+\omega .X)}, \sum \Vert U_{\tau , \omega }(t, x)\Vert _{C([0, t]:H^s({\Bbb R}^d))}&lt; \infty .\end{}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 from.An application to conical refraction where the characteristics have variable multiplicity is presented.
LA - eng
KW - interaction of high frequency solutions; geometric optics; crystal optics; oscillatory solutions; semilinear first order symmetric hyperbolic system; constant coefficients; Wiener algebra; application to conical refraction
UR - http://eudml.org/doc/75053
ER -

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Citations in EuDML Documents

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  1. Jean-Luc Joly, Guy Métivier, Jeffrey Rauch, Optique géométrique non linéaire et équations de Maxwell-Bloch
  2. Anatoli Babin, Alex Mahalov, Basil Nicolaenko, Fast singular oscillating limits and global regularity for the 3D primitive equations of geophysics
  3. J.-L. Joly, G. Métivier, J. Rauch, Coherent and focusing multidimensional nonlinear geometric optics
  4. Laure Saint-Raymond, Mathematical study of singular perturbation problems Applications to large-scale oceanography
  5. Anatoli Babin, Alex Mahalov, Basil Nicolaenko, Fast Singular Oscillating Limits and Global Regularity for the 3D Primitive Equations of Geophysics
  6. P. Donnat, J.-L. Joly, G. Métivier, J. Rauch, Diffractive nonlinear geometric optics
  7. Mark Williams, Boundary layers and glancing blow-up in nonlinear geometric optics
  8. J. L. Joly, G. Métivier, J. Rauch, Compacité par compensation trilinéaire et optique géométrique non linéaire

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