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Coherent and focusing multidimensional nonlinear geometric optics. (English) Zbl 0836.35087

A symmetric-hyperbolic system with weak nonlinear and semilinear terms \[ \partial_t U + \sum A_j (t,x, \varepsilon U) \partial_j U = F(t,x, \varepsilon U,U),\;x \in \mathbb{R}^n \] is considered. The main goal is to construct the first terms of the asymptotics of oscillatory solution of the form \[ U(t,x, \varepsilon) = U_0 (t,x) + \varepsilon U_1 (t,x, \Phi (t,x)/ \varepsilon) + o (\varepsilon),\;\varepsilon \to 0 \] with finite number of the fast phases \(\Phi = \{\varphi_1, \dots, \varphi_m\}\). As usually the phase functions \(\varphi_j\) are determined from the eikonal equation \(\partial_t \varphi + \sum A_j (t,x,0) \partial_j \varphi = 0\). The main object which appears in construction is the linear span of the \(\varphi_j\). The problem is solved under a very strong assumption named coherence property in some domain \(\Omega\). That is each nonzero function from the linear span of the \(\varphi_j\) either satisfy the eikonal equation identically or does not satisfies it at every point \((t,x) \in \Omega\).
Reviewer: L.Kalyakin (Ufa)

MSC:

35L60 First-order nonlinear hyperbolic equations
35B25 Singular perturbations in context of PDEs
35C20 Asymptotic expansions of solutions to PDEs
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References:

[1] C. CHEVERRY , Oscillations de faible amplitude pour les systèmes 2 \times 2 de lois de conservations , preprint Rennes, 1993 . Zbl 0852.35093 · Zbl 0852.35093
[2] J.-M. DELORT , Oscillations semi-linéaires multiphasées compatibles en dimension 2 et 3 d’espace (Comm. in Partial Diff. Equ., Vol. 16, 1991 , pp. 845-872). MR 92g:35138 | Zbl 0736.35001 · Zbl 0736.35001 · doi:10.1080/03605309108820781
[3] R. DIPERNA and A. MAJDA , The validity of geometric optics for weak solutions (Comm. in Math. Phys., Vol. 98, 1985 , pp. 313-347). Article | MR 87e:35057 | Zbl 0582.35081 · Zbl 0582.35081 · doi:10.1007/BF01205786
[4] O. GUES , Développements asymptotiques de solutions exactes de systèmes hyperboliques quasi-linéaires (Asymptotic Analysis, Vol. 6, 1993 , pp. 675-678). MR 94b:35067 | Zbl 0780.35017 · Zbl 0780.35017
[5] O. GUES , Ondes multidimensionnelles \epsilon -statifiées et oscillations (Duke Math. J., Vol. 68, 1992 , pp. 401-446). Article | MR 94a:35011 | Zbl 0837.35086 · Zbl 0837.35086 · doi:10.1215/S0012-7094-92-06816-5
[6] J. HUNTER , A. MAJDA and R. ROSALES , Resonantly Interacting Weakly Nonlinear Hyperbolic Waves II : Several Space Variables (Stud. Appl. Math., Vol. 75, 1986 , pp. 187-226). MR 89h:35199 | Zbl 0657.35084 · Zbl 0657.35084
[7] J.-L. JOLY , G. METIVIER and J. RAUCH , Resonant One Dimensional Nonlinear Geometric Optics (to appear in J. Funct. Analysis, Vol. 114, No. 1, 1993 , pp. 106-231). MR 94i:35118 | Zbl 0851.35023 · Zbl 0851.35023 · doi:10.1006/jfan.1993.1065
[8] J.-L. JOLY , G. METIVIER and J. RAUCH , Formal and Rigorous Nonlinear High Frequency Hyperbolic Waves , in Proceedings of Varenna Conference on Nonlinear Hyperbolic Equations and Field Theory, June 1990 , M. K. MURPTY, S. SPAGNOLO eds. (Pitman Research Note in Math. Series, 1992 , 121, p. 143). Zbl 0824.35077 · Zbl 0824.35077
[9] J.-L. JOLY , G. METIVIER and J. RAUCH , Remarques sur l’optique géométrique non linéaire multidimensionnelle , in Séminaire Équations aux dérivées partielles de l’École Polytechnique 1990 - 1991 , Exposé n^\circ 1. Numdam | Zbl 0749.35055 · Zbl 0749.35055
[10] J.-L. JOLY , G. METIVIER and J. RAUCH , Coherent Nonlinear Waves and the Wiener Algebra (Ann. Inst. Fourier, t. 44, 1994 , pp. 167-196). Numdam | MR 95c:35163 | Zbl 0791.35019 · Zbl 0791.35019 · doi:10.5802/aif.1393
[11] J.-L. JOLY , G. METIVIER and J. RAUCH , Generic rigorous asymptotic expansions for weakly nonlinear multidimensional oscillary waves (to appear in Duke Math. J., Vol. 70, No. 2, 1993 , pp. 373-404). Article | MR 94c:35048 | Zbl 0815.35066 · Zbl 0815.35066 · doi:10.1215/S0012-7094-93-07007-X
[12] J.-L. JOLY , G. METIVIER and J. RAUCH , Dense oscillations for the compressible 2-D Euler equations , in Non Linear Partial Differential Equations, Séminaire Collège de France 1992 - 1993 , Pitmann Publ. · Zbl 0920.35112
[13] J.-L. JOLY and J. RAUCH , Justification of Multidimensional Single Phase Semilinear Geometric Optics (Trans. Amer. Math. Soc., Vol. 330, 1992 , pp. 599-625). MR 92f:35040 | Zbl 0771.35010 · Zbl 0771.35010 · doi:10.2307/2153924
[14] J.-L. JOLY and J. RAUCH , Nonlinear Resonance can Create Dense Oscillations , in Microlocal Analysis and Nonlinear Waves, M. BEALS, R. MELROSE and J. B. RAUCH eds. (IMA Volumes in Math. and its Application, Vol. 30, Springer-Verlag). MR 92k:35180 | Zbl 0794.35098 · Zbl 0794.35098
[15] L. A. KALYAKIN , Long Waves Asymptotics of Solutions of Nonlinear Systems of Equations with Dispersion (Dokl. An SSSR, Vol. 288, n^\circ 4, 1986 , and Soviet Math. Dokl., Vol. 33, n^\circ 3, 1986 , pp. 769-774). MR 87j:35073 | Zbl 0629.35015 · Zbl 0629.35015
[16] L. A. KALYAKIN , Asymptotic Decay of a One Dimensional Wave Packet in a Nonlinear Dispersive Medium (Math. Sbornik Translation, Vol. 60, n^\circ 2, 1988 , pp. 457-484). MR 88h:35069 | Zbl 0699.35135 · Zbl 0699.35135 · doi:10.1070/SM1988v060n02ABEH003181
[17] L. A. KALYAKIN , Long Waves Asymptotics. Integrable Equations as Asymptotic Limit of Nonlinear Systems (Russian Mth. Surveys, Vol. 44, n^\circ 1, 1989 , pp. 3-42). MR 90g:58131 | Zbl 0683.35082 · Zbl 0683.35082 · doi:10.1070/RM1989v044n01ABEH002013
[18] J. KELLER , On solutions of nonlinear waves operators (Comm. Pure App. Math., Vol. 10, 1957 , pp. 523-530). MR 20 #3371 | Zbl 0090.31802 · Zbl 0090.31802 · doi:10.1002/cpa.3160100404
[19] P. LAX , Shock Waves and Entropy , in Contribution to Nonlinear Analysis, ZARANTONELLO ed., Academic Press, NY, 1971 , pp. 603-634. MR 52 #14677 | Zbl 0268.35014 · Zbl 0268.35014
[20] P. LAX , Asymptotic solutions of oscillatory initial value problems (Duke Math. J., Vol. 24, 1957 , pp. 627-645). Article | MR 20 #4096 | Zbl 0083.31801 · Zbl 0083.31801 · doi:10.1215/S0012-7094-57-02471-7
[21] A. MAJDA and R. ROSALES , Resonantly Interacting Weakly Nonlinear Hyperbolic Waves I : a Single Space Variables Stud. Appl. Math., Vol. 71, 1984 , pp. 149-179). MR 86e:35089 | Zbl 0572.76066 · Zbl 0572.76066
[22] J. RAUCH and M. REED , Striated solutions of semilinear twospeed wave equations (Indiana Univ. Math. J., Vol. 34, 1985 , pp. 337-353). MR 86m:35111 | Zbl 0559.35053 · Zbl 0559.35053 · doi:10.1512/iumj.1985.34.34020
[23] S. SCHOCHET , Fast singular limits of hyperbolic partial differential equations (to appear in J. Diff. Equ.). Zbl 0838.35071 · Zbl 0838.35071 · doi:10.1006/jdeq.1994.1157
[24] S. SCHOCHET , Resonant nonlinear geometric optics for weak solutions of conservations laws , preprint 1992 . · Zbl 0856.35080
[25] M. SABLÉ TOUGERON , Justification de l’optique géométrique faiblement non linéaire pour le problème mixte : cas des concentrations , preprint, 1993 . · Zbl 0861.35059
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