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

show all references

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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