Advanced Search
Article Contents
Article Contents

On some geometry of propagation in diffractive time scales

Abstract Related Papers Cited by
  • In this article, we develop a non linear geometric optics which presents the two main following features. It is valid in diffractive times and it extends the classical approaches [7, 17, 18, 24] to the case of fast variable coefficients. In this context, we can show that the energy is transported along the rays associated with a non usual long-time hamiltonian. Our analysis needs structural assumptions and initial data suitably polrarized to be implemented. All the required conditions are met concerning a current model [2, 3, 8, 9, 10, 11, 19, 21] arising in fluid mechanics, which was the original motivation of our work. As a by product, we get results complementary to the litterature concerning the propagation of the Rossby waves which play a part in the description of large oceanic currents, like Gulf stream or Kuroshio.
    Mathematics Subject Classification: Primary: 35L, 35Q; Secondary: 76, 37.


    \begin{equation} \\ \end{equation}
  • [1]

    C. Cheverry, Justification de l'optique géométrique non linéaire pour un système de lois de conservation, Duke Math. J., 87 (1997), 213-263.doi: 10.1215/S0012-7094-97-08710-X.


    C. Cheverry, I. Gallagher, T. Paul and L. Saint-RaymondSemiclassical and spectral analysis of oceanic waves, To appear in Duke Math. J., arXiv:1005.1146.


    C. Cheverry, I. Gallagher, T. Paul and L. Saint-Raymond, Trapping Rossby waves, C. R. Math. Acad. Sci. Paris, 347 (2009), 879-884.


    C. Cheverry, O. Guès and G. Métivier, Oscillations fortes sur un champ linéairement dégénéré, Ann. Sci. Ècole Norm. Sup., 36 (2003), 691-745.


    Y. Choquet-Bruhat, Ondes asymptotiques et approchées pour des systèmes d'équations aux dérivées partielles non linéaires, J. Math. Pures Appl., 48 (1969), 117-158.


    P. Donnat, J.-L. Joly, G. Metivier and J. Rauch, Diffractive nonlinear geometric optics, In "Séminaire sur les Èquations aux Dérivées Partielles," 1995-1996, Exp. No. XVII, 25 pp., Ècole Polytech., Palaiseau, 1996.


    E. Dumas, Periodic multiphase nonlinear diffractive optics with curved phases, Indiana Univ. Math. J., 52 (2003), 769-810.


    A. Dutrifoy and A. Majda, The dynamics of equatorial long waves: A singular limit with fast variable coefficients, Commun. Math. Sci., 4 (2006).


    A. Dutrifoy, A. Majda and S. Schochet, A simple justification of the singular limit for equatorial shallow-water dynamics, Comm. Pure Appl. Math., 62 (2009), 322-333.doi: 10.1002/cpa.20248.


    I. Gallagher and L. Saint-Raymond, Mathematical study of the betaplane model: Equatorial waves and convergence results, Mém. Soc. Math. Fr., 107 (2007), v+116 pp.


    H. P. Greenspan, "The Theory of Rotating Fluids," Cambridge Monographs on Mechanics and Applied Mathematics, 1980, 328 pp.


    O. Guès, Ondes multidimensionnelles $\epsilon$-stratifiées et oscillations, Duke Math. J., 68 (1992), 401-446.doi: 10.1215/S0012-7094-92-06816-5.


    O. Guès, Développement asymptotique de solutions exactes de systèmes hyperboliques quasilinéaires, Asymptotic Anal., 6 (1993), 241-269.


    J. K. Hunter, A. Majda and R. Rosales, Resonantly interacting, weakly nonlinear hyperbolic waves. II. Several space variables, Stud. Appl. Math., 75 (1986), 187-226.


    J.-L. Joly, G. Métivier and J. Rauch, Coherent and focusing multidimensional nonlinear geometric optics, Ann. Sci. École Norm. Sup., 28 (1995), 51-113.


    J.-L. Joly, G. Métivier and J. Rauch, Nonlinear oscillations beyond caustics, Comm. Pure Appl. Math., 49 (1996), 443-527.doi: 10.1002/(SICI)1097-0312(199605)49:5<443::AID-CPA1>3.0.CO;2-B.


    J.-L. Joly, G. Métivier and J. Rauch, Transparent nonlinear geometric optics and Maxwell-Bloch equations, J. Differential Equations, 166 (2000), 175-250.doi: 10.1006/jdeq.2000.3794.


    D. Lannes and J. Rauch, Validity of nonlinear geometric optics with times growing logarithmically, Proc. Amer. Math. Soc., 129 (2001), 1087-1096.doi: 10.1090/S0002-9939-00-05845-7.


    A. Majda, "Introduction to PDEs and Waves for the Atmosphere and Ocean," Courant Lecture Notes in Mathematics, 9, Providence, RI, American Mathematical Society (AMS), New York, NY, Courant Institute of Mathematical Sciences, 1996, 234 p.


    T. Paul, Échelles de temps pour l'évolution quantique à petite constante de Planck (French) [Time scales of a quantum evolution with small Planck constant], In "Séminaire sur les Équations aux Dérivées Partielles," 2007-2008, Exp. No. IV, 21 pp., École Polytech., Palaiseau, 2009.


    J. Pedlosky, "Ocean Circulation Theory," Springer, 1996.


    D. Sanchez, Long waves in ferromagnetic media, Khokhlov-Zabolotskaya equation, J. Differential Equations, 210 (2005), 263-289.doi: 10.1016/j.jde.2004.08.017.


    R. Sentis, Mathematical models for laser-plasma interaction, M2AN Math. Model. Numer. Anal., 39 (2005), 275-318.doi: 10.1051/m2an:2005014.


    B. Texier, The short-wave limit for nonlinear, symmetric, hyperbolic systems, Adv. Differential Equations, 9 (2004), 1-52.

  • 加载中

Article Metrics

HTML views() PDF downloads(78) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint