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 description
where the profile is almost periodic in .
The main novelty in the analysis is the space of profiles which have the form
Thus, 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.
On étudie les solutions oscillantes de systèmes semi linéaires du premier ordre , avec hyperbolique symétrique et fonction analytique réelle de ses arguments. La principale nouveauté est l’étude de problèmes multidimensionnels sous l’hypothèse naturelle de cohérence des phases, pour des profils presque périodiques non nécessairement quasi-périodiques. Par exemple, lorsque est à coefficients constants et les phases sont linéaires, on construit des solutions qui possèdent une asymptotique haute fréquence de la forme
L’analyse se fait avec des profils dans l’algèbre de Wiener des fonctions presque périodiques en .
Les profils sont uniquement déterminés à partir de leurs données initiales par le système habituel des équations de l’optique géométrique non linéaire.
L’analyse s’applique au cas des systèmes à caractéristiques de multiplicité variable, et permet de traiter l’exemple de la réfraction conique.
@article{AIF_1994__44_1_167_0, author = {Joly, J.-L. and M\'etivier, G. and Rauch, J.}, title = {Coherent nonlinear waves and the {Wiener} algebra}, journal = {Annales de l'Institut Fourier}, pages = {167--196}, publisher = {Institut Fourier}, address = {Grenoble}, volume = {44}, number = {1}, year = {1994}, doi = {10.5802/aif.1393}, mrnumber = {95c:35163}, zbl = {0791.35019}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.1393/} }
TY - JOUR AU - Joly, J.-L. AU - Métivier, G. AU - Rauch, J. TI - Coherent nonlinear waves and the Wiener algebra JO - Annales de l'Institut Fourier PY - 1994 SP - 167 EP - 196 VL - 44 IS - 1 PB - Institut Fourier PP - Grenoble UR - http://archive.numdam.org/articles/10.5802/aif.1393/ DO - 10.5802/aif.1393 LA - en ID - AIF_1994__44_1_167_0 ER -
%0 Journal Article %A Joly, J.-L. %A Métivier, G. %A Rauch, J. %T Coherent nonlinear waves and the Wiener algebra %J Annales de l'Institut Fourier %D 1994 %P 167-196 %V 44 %N 1 %I Institut Fourier %C Grenoble %U http://archive.numdam.org/articles/10.5802/aif.1393/ %R 10.5802/aif.1393 %G en %F AIF_1994__44_1_167_0
Joly, J.-L.; Métivier, G.; Rauch, J. Coherent nonlinear waves and the Wiener algebra. Annales de l'Institut Fourier, Volume 44 (1994) no. 1, pp. 167-196. doi : 10.5802/aif.1393. http://archive.numdam.org/articles/10.5802/aif.1393/
[BW]Principles of optics, 4th ed., Pergamon Press, Oxford, 1970.
and ,[CB]Ondes asymptotiques et approchées pour systèmes d'équations aux dérivées partielles non linéaires, J. Math. Pure Appl., 48 (1969), 117-158. | MR | Zbl
,[C]Methods of mathematical physics, vol. II, Interscience Publishers, 1962. | Zbl
,[D]Oscillations semi-linéaires multiphasées compatibles en dimension 2 et 3 d'espace, J. Diff. Eq., 1991. | Zbl
,[DM]The validity of geometric optics for weak solutions of conservation laws, Comm. Math. Phys., 98 (1985), 313-347. | MR | Zbl
and ,[G1]Développements asymptotiques de solutions exactes de systèmes hyperboliques quasilinéaires, Asymptotic Anal., 1993. | Zbl
,[G2]Ondes multidimensionnelles epsilon stratifiées et oscillations, Duke Math. J., 1992. | MR | Zbl
,[Hor]The analysis of linear partial differential operators, vol. 1, Springer-Verlag, 1991.
,[Hou]Homogenization for semilinear hyperbolic systems with oscillatory data, Comm. Pure Appl. Math., 41 (1988), 471-495. | MR | Zbl
,[HK]Weakly nonlinear high frequency waves, Comm. Pure Appl. Math., 36 (1983), 547-569. | MR | Zbl
and ,[HMR]Resonantly interacting weakly nonlinear hyperbolic waves II : several space variables, Stud. Appl. Math., 75 (1986), 187-226. | MR | Zbl
, and ,[JMR1] Resonant one dimensional non linear geometric optics, J. Funct. Anal., 114 (1993), 106-231. | MR | Zbl
, and ,[JMR2]Rigorous resonant 1−d nonlinear geometric optics, in Journées Équations aux Dérivées Partielles, St Jean de Monts, juin 1990, Publ. de l'École Polytechnique, Palaiseau. | EuDML | Numdam | MR | Zbl
, and ,[JMR3]Formal and rigorous nonlinear high frequency hyperbolic waves, in Nonlinear Hyperbolic Equations and Field Theory, eds. M.K.V. Murthy and S. Spagnolo, Pitmann Research Notes in Mathematics #253, 1992, 121-144. | MR | Zbl
, and ,[JMR4]Remarques sur l'optique géométrique non linéaire multidimensionnelle, Séminaire Équations aux Dérivées Partielles, École Polytechnique, exposé n° 1, 1990-1991. | EuDML | Numdam | MR | Zbl
, and ,[JMR5]Coherent and focussing multidimensional nonlinear geometric optics, Annales de l'École Normale Supérieure, to appear. | EuDML | Numdam | MR | Zbl
, and ,[JMR6]Generic rigorous asymptotic expansions for weakly nonlinear multidimensional oscillatory waves, Duke Math. J., 70 (1993), 373-404. | MR | Zbl
, and ,[JMR7]Nonlinear geometric optics with an oscillating plane, preprint.
, and ,[J]Sur la propagation des oscillations semi-linéaires en dimension 1 d'espace, C.R. Acad. Sc. Paris, t. 296 (1983). | MR | Zbl
,[JR1]Ondes oscillantes semi-linéaires en 1−d, in Journées Équations aux Dérivées Partielles, St Jean de Monts, juin 1986, Publ. de l'École Polytechnique, Palaiseau. | EuDML | Numdam | MR | Zbl
and ,[JR2]Ondes oscillantes semi-linéaires à hautes fréquences, in Recent Developments in Hyperbolic Equations, (L. Cattabriga, F. Colombini, M. Murthy, S. Spagnolo, eds.), Pitman Research Notes in Math., 183 (1988), 103-115. | MR | Zbl
and ,[JR3]High frequency semilinear oscillations, in Wave Motion : Theory, Modelling and Computation (A.-J. Chorin and A.-J. Majda, eds.), Springer-Verlag (1987), 202-217. | MR | Zbl
and ,[JR4]Nonlinear resonance can create dense oscillations, in Microlocal Analysis and Nonlinear Waves, M. Beals, R. Melrose and J. Rauch eds.), Springer-Verlag (1991), 113-123. | MR | Zbl
and ,[JR5]Justification of multidimensional single phase semilinear geometric optics, Trans. Amer. Math. Soc., 330 (1992), 599-625. | MR | Zbl
and ,[KAl]Long wave asymptotics, integrable equations as asymptotic limit of nonlinear systems, Russian Math. Surveys, vol. 44, n° 1 (1989), 3-42. | MR | Zbl
,[Kat]An introduction to harmonic analysis 2d ed., Dover Publ., 1976. | MR | Zbl
,[Kl]The null condition and global existence to nonlinear wave equations, Springer Lectures in Applied Mathematics, 23 (1986), 293-326. | MR | Zbl
,[Kr]Über sachgemässe cauchyprobleme, Math. Scand., 13 (1963), 109-128. | EuDML | MR | Zbl
,[La]Asymptotic solutions of oscillatory initial value problems, Duke Math. J., 24 (1957), 627-646. | MR | Zbl
,[Lu]Conical refraction in crystal optics and hydromagnetics, Comm. Pure Appl. Math., XIV (1961), 113-124. | MR | Zbl
,[MR]Resonantly interacting weakly nonlinear hyperbolic waves I : a single space variable, Stud. Appl. Math., 71 (1984), 149-179. | MR | Zbl
and ,[MRS]A canonical system of integrodifferential equations in nonlinear acoustics, Stud. Appl. Math., 79 (1988), 205-262. | MR | Zbl
, and ,[MPT]Weak limits of semilinear hyperbolic systems with oscillating data, Lecture Notes in Physics 230, Springer-Verlag (1985), 277-289. | MR | Zbl
, and ,[MU]Microlocal structure of involutive conical refraction, Duke Math. J., 46 (1979), 118-133. | MR | Zbl
and ,[S]Fast singular limits of hyperbolic partial differential equations, J. Diff. Eq., to appear. | MR | Zbl
,[Tar]Solutions oscillantes des équations de Carleman, Séminaire Goulaouic-Meyer-Schwartz, 1983. | Numdam | Zbl
,[Tay]Pseudodifferential operators, Princeton Mathematics Series #34, Princeton University Press, Princeton N.J., 1981. | MR | Zbl
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