# American Institute of Mathematical Sciences

February  2012, 32(2): 499-538. doi: 10.3934/dcds.2012.32.499

## On some geometry of propagation in diffractive time scales

 1 UMR6625, Université Rennes 1, Campus de Beaulieu, 263 avenue du Général Leclerc CS 74205, 35042 Rennes, France 2 UMR7640, Centre de Mathmatiques Laurent Schwartz, École polytechnique, France

Received  September 2010 Revised  December 2010 Published  September 2011

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.
Citation: Christophe Cheverry, Thierry Paul. On some geometry of propagation in diffractive time scales. Discrete & Continuous Dynamical Systems - A, 2012, 32 (2) : 499-538. doi: 10.3934/dcds.2012.32.499
##### References:
 [1] C. Cheverry, Justification de l'optique géométrique non linéaire pour un système de lois de conservation,, Duke Math. J., 87 (1997), 213. doi: 10.1215/S0012-7094-97-08710-X. Google Scholar [2] C. Cheverry, I. Gallagher, T. Paul and L. Saint-Raymond, Semiclassical and spectral analysis of oceanic waves,, To appear in Duke Math. J., (). Google Scholar [3] C. Cheverry, I. Gallagher, T. Paul and L. Saint-Raymond, Trapping Rossby waves,, C. R. Math. Acad. Sci. Paris, 347 (2009), 879. Google Scholar [4] C. Cheverry, O. Guès and G. Métivier, Oscillations fortes sur un champ linéairement dégénéré,, Ann. Sci. Ècole Norm. Sup., 36 (2003), 691. Google Scholar [5] Y. Choquet-Bruhat, Ondes asymptotiques et approchées pour des systèmes d'équations aux dérivées partielles non linéaires,, J. Math. Pures Appl., 48 (1969), 117. Google Scholar [6] P. Donnat, J.-L. Joly, G. Metivier and J. Rauch, Diffractive nonlinear geometric optics,, In, (1996), 1995. Google Scholar [7] E. Dumas, Periodic multiphase nonlinear diffractive optics with curved phases,, Indiana Univ. Math. J., 52 (2003), 769. Google Scholar [8] A. Dutrifoy and A. Majda, The dynamics of equatorial long waves: A singular limit with fast variable coefficients,, Commun. Math. Sci., 4 (2006). Google Scholar [9] 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. doi: 10.1002/cpa.20248. Google Scholar [10] I. Gallagher and L. Saint-Raymond, Mathematical study of the betaplane model: Equatorial waves and convergence results,, Mém. Soc. Math. Fr., 107 (2007). Google Scholar [11] H. P. Greenspan, "The Theory of Rotating Fluids,", Cambridge Monographs on Mechanics and Applied Mathematics, (1980). Google Scholar [12] O. Guès, Ondes multidimensionnelles $\epsilon$-stratifiées et oscillations,, Duke Math. J., 68 (1992), 401. doi: 10.1215/S0012-7094-92-06816-5. Google Scholar [13] O. Guès, Développement asymptotique de solutions exactes de systèmes hyperboliques quasilinéaires,, Asymptotic Anal., 6 (1993), 241. Google Scholar [14] J. K. Hunter, A. Majda and R. Rosales, Resonantly interacting, weakly nonlinear hyperbolic waves. II. Several space variables,, Stud. Appl. Math., 75 (1986), 187. Google Scholar [15] J.-L. Joly, G. Métivier and J. Rauch, Coherent and focusing multidimensional nonlinear geometric optics,, Ann. Sci. École Norm. Sup., 28 (1995), 51. Google Scholar [16] J.-L. Joly, G. Métivier and J. Rauch, Nonlinear oscillations beyond caustics,, Comm. Pure Appl. Math., 49 (1996), 443. doi: 10.1002/(SICI)1097-0312(199605)49:5<443::AID-CPA1>3.0.CO;2-B. Google Scholar [17] J.-L. Joly, G. Métivier and J. Rauch, Transparent nonlinear geometric optics and Maxwell-Bloch equations,, J. Differential Equations, 166 (2000), 175. doi: 10.1006/jdeq.2000.3794. Google Scholar [18] D. Lannes and J. Rauch, Validity of nonlinear geometric optics with times growing logarithmically,, Proc. Amer. Math. Soc., 129 (2001), 1087. doi: 10.1090/S0002-9939-00-05845-7. Google Scholar [19] A. Majda, "Introduction to PDEs and Waves for the Atmosphere and Ocean,", Courant Lecture Notes in Mathematics, 9 (1996). Google Scholar [20] 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, (2009), 2007. Google Scholar [21] J. Pedlosky, "Ocean Circulation Theory,", Springer, (1996). Google Scholar [22] D. Sanchez, Long waves in ferromagnetic media, Khokhlov-Zabolotskaya equation,, J. Differential Equations, 210 (2005), 263. doi: 10.1016/j.jde.2004.08.017. Google Scholar [23] R. Sentis, Mathematical models for laser-plasma interaction,, M2AN Math. Model. Numer. Anal., 39 (2005), 275. doi: 10.1051/m2an:2005014. Google Scholar [24] B. Texier, The short-wave limit for nonlinear, symmetric, hyperbolic systems,, Adv. Differential Equations, 9 (2004), 1. Google Scholar

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##### References:
 [1] C. Cheverry, Justification de l'optique géométrique non linéaire pour un système de lois de conservation,, Duke Math. J., 87 (1997), 213. doi: 10.1215/S0012-7094-97-08710-X. Google Scholar [2] C. Cheverry, I. Gallagher, T. Paul and L. Saint-Raymond, Semiclassical and spectral analysis of oceanic waves,, To appear in Duke Math. J., (). Google Scholar [3] C. Cheverry, I. Gallagher, T. Paul and L. Saint-Raymond, Trapping Rossby waves,, C. R. Math. Acad. Sci. Paris, 347 (2009), 879. Google Scholar [4] C. Cheverry, O. Guès and G. Métivier, Oscillations fortes sur un champ linéairement dégénéré,, Ann. Sci. Ècole Norm. Sup., 36 (2003), 691. Google Scholar [5] Y. Choquet-Bruhat, Ondes asymptotiques et approchées pour des systèmes d'équations aux dérivées partielles non linéaires,, J. Math. Pures Appl., 48 (1969), 117. Google Scholar [6] P. Donnat, J.-L. Joly, G. Metivier and J. Rauch, Diffractive nonlinear geometric optics,, In, (1996), 1995. Google Scholar [7] E. Dumas, Periodic multiphase nonlinear diffractive optics with curved phases,, Indiana Univ. Math. J., 52 (2003), 769. Google Scholar [8] A. Dutrifoy and A. Majda, The dynamics of equatorial long waves: A singular limit with fast variable coefficients,, Commun. Math. Sci., 4 (2006). Google Scholar [9] 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. doi: 10.1002/cpa.20248. Google Scholar [10] I. Gallagher and L. Saint-Raymond, Mathematical study of the betaplane model: Equatorial waves and convergence results,, Mém. Soc. Math. Fr., 107 (2007). Google Scholar [11] H. P. Greenspan, "The Theory of Rotating Fluids,", Cambridge Monographs on Mechanics and Applied Mathematics, (1980). Google Scholar [12] O. Guès, Ondes multidimensionnelles $\epsilon$-stratifiées et oscillations,, Duke Math. J., 68 (1992), 401. doi: 10.1215/S0012-7094-92-06816-5. Google Scholar [13] O. Guès, Développement asymptotique de solutions exactes de systèmes hyperboliques quasilinéaires,, Asymptotic Anal., 6 (1993), 241. Google Scholar [14] J. K. Hunter, A. Majda and R. Rosales, Resonantly interacting, weakly nonlinear hyperbolic waves. II. Several space variables,, Stud. Appl. Math., 75 (1986), 187. Google Scholar [15] J.-L. Joly, G. Métivier and J. Rauch, Coherent and focusing multidimensional nonlinear geometric optics,, Ann. Sci. École Norm. Sup., 28 (1995), 51. Google Scholar [16] J.-L. Joly, G. Métivier and J. Rauch, Nonlinear oscillations beyond caustics,, Comm. Pure Appl. Math., 49 (1996), 443. doi: 10.1002/(SICI)1097-0312(199605)49:5<443::AID-CPA1>3.0.CO;2-B. Google Scholar [17] J.-L. Joly, G. Métivier and J. Rauch, Transparent nonlinear geometric optics and Maxwell-Bloch equations,, J. Differential Equations, 166 (2000), 175. doi: 10.1006/jdeq.2000.3794. Google Scholar [18] D. Lannes and J. Rauch, Validity of nonlinear geometric optics with times growing logarithmically,, Proc. Amer. Math. Soc., 129 (2001), 1087. doi: 10.1090/S0002-9939-00-05845-7. Google Scholar [19] A. Majda, "Introduction to PDEs and Waves for the Atmosphere and Ocean,", Courant Lecture Notes in Mathematics, 9 (1996). Google Scholar [20] 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, (2009), 2007. Google Scholar [21] J. Pedlosky, "Ocean Circulation Theory,", Springer, (1996). Google Scholar [22] D. Sanchez, Long waves in ferromagnetic media, Khokhlov-Zabolotskaya equation,, J. Differential Equations, 210 (2005), 263. doi: 10.1016/j.jde.2004.08.017. Google Scholar [23] R. Sentis, Mathematical models for laser-plasma interaction,, M2AN Math. Model. Numer. Anal., 39 (2005), 275. doi: 10.1051/m2an:2005014. Google Scholar [24] B. Texier, The short-wave limit for nonlinear, symmetric, hyperbolic systems,, Adv. Differential Equations, 9 (2004), 1. Google Scholar
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