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 L u = f ( t , x , u , 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 ϵ = U ( t , x , t / ϵ , x / ϵ ) + 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 = ∑ τ , ω ∈ ℝ 1 + d U τ , ω ( t , x ) e i ( τ T + ω . X ) , ∑ ∥ U τ , ω ( t , x ) ∥ C ( [ 0 , t ] : H s ( ℝ d ) ) < ∞ . 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 ϵ 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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