Alain BACHELOT
Université de Bordeaux


Adresse
Institut de Mathématiques de Bordeaux
351, cours de la Libération
F-33405 TALENCE cedex

bâtiment A33, bureau 283

Tél : (33 5) / (05) 40 00 60 63
Fax. (33 5) / (05) 40 00 26 26
E-mail :
alain.bachelot@u-bordeaux.fr

Pour plus de détails : Pour visiter l'Université de Bordeaux, l'UF mathématiques et interactions, la Bibliothèque, l'Institut de Mathématiques, l'Equipe EDP et physique mathématique, ou revoir Newton.

Dernière mise à jour, 6 juin 2018.

  • Président de la Commission de spécialistes 26° section (1991-1994)
  • Directeur de l'Ecole doctorale de mathématiques et informatique de Bordeaux (1995-1998)
  • Membre élu du Conseil National des Universités, membre du bureau (1996-1998)
  • Membre nommé au Conseil National des Universités (1999-2003)
  • Directeur de l'UMR CNRS 5466 de mathématiques appliquées (1999-2002) et du LRC CEA M03 (2000-2002)
  • Membre (2002-2007) et Secrétaire (2003-2007) du Conseil Scientifique de la SMF
  • Membre du Comité pour l'égalité professionnelle hommes/femmes - Ministère de la Recherche (2006-2010)
  • Président de la Commission Consultative 26° section (2008-2010)
  • Responsable de la licence d'ingénierie mathématique (2007-2010)
  • Responsable du Master MIMSE (2010-2015)
  • Responsable de l'équipe "EDP et physique mathématique" (2010-2014)
  • Thèmes de recherche


     
      PUBLICATIONS



    Comptes-rendus de l'Académie des Sciences de Paris / Academy of Sciences of Paris


    A. BACHELOT. Problème inverse de diffusion non linéaire. C. R. Acad. Sc. Paris, Série I, 293 : 121-124, 1981.
     
    A. BACHELOT. Problème de Cauchy pour des systèmes de Klein-Gordon-Schrödinger. C. R. Acad. Sc. Paris, Série I, 296 : 525-528, 1983.
     
    A. BACHELOT, B. HANOUZET. Applications bilinéaires compatibles avec un système différentiel à coefficients variables. C. R. Acad. Sc. Paris, Série I, 299 : 543-546, 1984.
     
    A. BACHELOT. Equipartition de l'énergie pour les systèmes hyperboliques et formes compatibles. C. R. Acad. Sc. Paris, Série I, 301 : 573-576, 1985.
     
    A. BACHELOT, V. PETKOV. Existence de l'opérateur de diffusion pour l'équation des ondes avec un potentiel périodique en temps. C. R. Acad. Sc. Paris, Série I, 303 : 671-673, 1986.
     
    A. BACHELOT. Opérateur de diffraction pour le système de Maxwell en métrique de Schwarzschild. C. R. Acad. Sc. Paris, Série I, 312 : 93-96, 1991.
     
    A. BACHELOT, A. PUJOLS. Equations intégrales espace-temps pour le système de Maxwell. C. R. Acad. Sc. Paris, Série I, 314 : 639-644, 1992.
     
    A. BACHELOT, A. MOTET-BACHELOT. Les pôles de résonance de la métrique de Schwarzschild. C. R. Acad. Sc. Paris, Série I, 316 : 795-798, 1993.
     
    A. BACHELOT, J-P. NICOLAS. Equation non linéaire de Klein-Gordon dans des métriques de type Schwarzschild. C. R. Acad. Sc. Paris, Série I, 316 : 1047-1050, 1993.
     
    A. BACHELOT. Opérateur de diffusion classique et quantique pour l'équation de Klein-Gordon en métrique de Schwarzschild. C. R. Acad. Sc. Paris, Série I, 319 : 41-44, 1994.
     
    A. BACHELOT. Diffusion d'un champ scalaire par un effondrement gravitationnel. C. R. Acad. Sc. Paris, Série I, 321 : 1329-1332, 1995.
     
    A. BACHELOT. La radiation Hawking à l'horizon d'un trou noir. C. R. Acad. Sc. Paris, Série I, 324 : 855-860, 1997.
     
    A. BACHELOT. L'effet Hawking. C. R. Acad. Sc. Paris, Série I, 325 : 1229-1234, 1997.
     
    A. BACHELOT. Création de fermions à l'horizon d'un trou noir chargé. C. R. Acad. Sc. Paris, Série I, 330 : 28-34, 2000.
     
    A. BACHELOT. Diffusion des ondes par une violation de la causalité. C. R. Acad. Sc. Paris, Série I, 333 : 1065-1068, 2001.
     
    A. BACHELOT, E. DARRIGRAND, K. MER-NKONGA. Coupling of a Multilevel fast Multipole and a Microlocal Discretization for the 3-D Integral Equations of Electromagnetism. C. R. Acad. Sc. Paris, Série I, 336 : 505-510, 2003.    

    A. BACHELOT. Paradoxe de Klein pour l'équation de Klein-Gordon chargée : superradiance et opérateur de diffusion. C. R. Acad. Sc. Paris, Série I, 339 : 345-350, 2004.
     
    A. BACHELOT. The Dirac Equation on the Anti-De-Sitter Universe. C. R. Acad. Sc. Paris, Série I, 345 : 435-440, 2007.
     
    A. BACHELOT. Scattering by a Minkowski Brane World. C. R. Acad. Sc. Paris, Série I,  347 : 1243-1248, 2009.

    A. BACHELOT. On the wave propagation in the anti-de Sitter cosmology. C. R. Math. Acad. Sci. Paris, Série I,  349 : 47-51, 2011.

    A. BACHELOT. New boundary conditions on the time-like conformal infinity of the anti-de Sitter universe. C. R. Math. Acad. Sci. Paris, Série I, 350 : 359-364, 2012.

    A. BACHELOT. Wave fluctuations near a De Sitter brane in an ti-de Sitter universe. C. R. Math. Acad. Sci. Paris, Série I, 354 : 19-25, 2016.


    Articles dans des revues à comité de lecture / Articles in Journal with referee
     
     

    A. BACHELOT. Problème de Cauchy pour des systèmes hyperboliques semi-linéaires. Ann. Inst. Henri Poincaré, Analyse non linéaire, 1(6) : 453-478, 1984.
     
    A. BACHELOT. Convergence dans $L^p(\BbbR^n+1)$ de la solution de l'équation de Klein-Gordon vers celle de l'équation des ondes. Ann. Fac. Sci. Toulouse, 8(1) : 37-60, 1986.
     
    A. BACHELOT. Equipartition de l'énergie pour les systèmes hyperboliques et formes compatibles. Ann. Inst. Henri Poincaré - Physique théorique, 46(1) : 45-76, 1987.
     
    A. BACHELOT, V. PETKOV. Existence des opérateurs d'ondes pour les systèmes hyperboliques avec un potentiel périodique en temps. Ann. Inst. Henri Poincaré - Physique théorique, 47(4) : 383-428, 1987.
     
    A. BACHELOT. Problème de Cauchy global pour des systèmes de Dirac-Klein-Gordon. Ann. Inst. Henri Poincaré - Physique théorique, 48(4) : 387-422, 1988.
     
    A. BACHELOT. Global existence of large amplitude solutions for non linear massless Dirac equations. Portugaliae Math., 46(5) : 455-473, 1989.
     
    A. BACHELOT. Gravitationnal Scattering of Electromagnetic Field by Schwarzschild Black Hole. Ann. Inst. Henri Poincaré - Physique théorique, 54(3) : 261-320, 1991.
     
    A. BACHELOT, A. MOTET-BACHELOT. Les résonances d'un trou noir de Schwarzschild. Ann. Inst. Henri Poincaré - Physique théorique, 59(1) : 3-68, 1993.
     
    A. BACHELOT.  Asymptotic Completeness for the Klein-Gordon Equation on the Schwarzschild Metric. Ann. Inst. Henri Poincaré - Physique théorique, 61(4) : 411-441, 1994.
     
    A. BACHELOT.  Scattering of Scalar Fields by Spherical Gravitational Collapse. J. Math. Pures Appl., 76(2) : 155-210, 1997.
     
    A. BACHELOT, A. PUJOLS. Décroissance de l'énergie locale d'une onde diffusée par un obstacle inhomogène. Rev. Roumaine Math. Pures et Appl., 41(7-8) : 451-459, 1997.
     
    A. BACHELOT. Quantum Vacuum Polarization at the Black-Hole Horizon. Ann. Inst. Henri Poincaré - Physique théorique, 67(2) : 181-222, 1997.
     
    A. BACHELOT. The Hawking Effect. Ann. Inst. Henri Poincaré - Physique théorique, 70(1) : 41-99, 1999.
     
    A. BACHELOT. Creation of Fermions at the Charged Black-Hole Horizon. Ann. Henri Poincaré, 1 : 1043-1095, 2000.
     
    A. BACHELOT, L. BOUNHOURE, A. PUJOLS. Couplage éléments finis-potentiels retardés pour la diffraction électromagnétique par un obstacle inhomogène. Numer. Math., 89 : 257-306, 2001.
     
    A. BACHELOT. Global properties of the wave equation on non globally hyperbolic manifolds. J. Math. Pures Appl., 81 : 35-65, 2002.
     
    A. BACHELOT. Superradiance and Scattering of the Charged Klein-Gordon Field by a Steplike Electrostatic Potential. J. Math. Pures Appl., 83 : 1179-1239, 2004.
     
    A. BACHELOT. Global Cauchy problem for semilinear hyperbolic systems with non-local nonlinearities. Applications to Dirac equations. J. Math. Pures Appl., 86 : 201-236, 2006.
     
    A. BACHELOT. Global waves with non positive energy in general relativity. Serdica Math. J., 34 : 127-154, 2008.
     
    A. BACHELOT. The Dirac System on the Anti-de Sitter Universe . Comm. Math. Phys. 283 : 127-167, 2008.
     
    A. BACHELOT. Wave Propagation and Scattering for the RS2 Brane Cosmology Model. J. Hyperbolic Differ. Equ., 6(4) : 809-861, 2009.
     
    A. BACHELOT.  The Klein-Gordon Equation in Anti-de Sitter Cosmology, J. Math. Pures Appl., 96 : 527-554, 2011.

    A. BACHELOT. New Dynamics in the Anti-de Sitter Universe AdS^5. Comm. Math. Phys., 320, 723-759, 2013.

    A. BACHELOT. On the Klein-Gordon equation near a De Sitter brane in an Anti-de Sitter bulk, J. Math. Pures Appl., 105, 165-197, 2016.

    A. BACHELOT. Waves in the Witten Bubble of Nothing and the Hawking Wormhole. Comm. Math. Phys., 351(2), 599–651, 2017. To read online

    A. BACHELOT-MOTET, A.BACHELOT. Waves on accelerating dodecahedral universes, Class. Quantum Gravity, 35(5), 055010 (39pp), 2017.



    Conférences publiées avec comité de lecture / Proceedings with referee



    A. BACHELOT. Inverse scattering problem for the nonlinear Klein-Gordon equation. In J-I. Diaz J. Herdandez C. Bardos, A. Damlamian, editor, Contributions to nonlinear partial differentials equations, volume 89 of Research Notes in Mathematics, pages 7-15. Pitman, 1983.

     

    A. BACHELOT. Problèmes de Cauchy pour des systèmes hyperboliques semi-linéaires. In Equations aux dérivées partielles, Saint Jean de Monts, pages VIII.1-VIII.5, 1983.
     

    A. BACHELOT, V. PETKOV. Existence de l'opérateur de diffusion pour l'équation des ondes avec un potentiel périodique en temps. In J-L. Lions, editor, Nonlinear partial differentials equations and their applications - Collège de France seminar, volume 181 of Research Notes in Mathematics, pages 13-27. Pitman, 1983.
     

    A. BACHELOT. Equipartition de l'énergie pour les systèmes hyperboliques et formes compatibles. In Equations aux dérivées partielles, Saint Jean de Monts, pages XIII.1-XIII.8, 1986.
     

    A. BACHELOT. Opérateurs de convolution définis à partir d'une forme quadratique. In Equations aux dérivées partielles, Saint Jean de Monts, pages XVI.1-XVI.8, 1982.

    A. BACHELOT. Solutions globales des systèmes de Dirac-Klein-Gordon. In Equations aux dérivées partielles, Saint Jean de Monts, pages XV.1-XV.10, 1987.

     

    A. BACHELOT. Global existence of large amplitude solutions for Dirac-Klein-Gordon systems in Minkowski space. In Non linear Hyperbolic Problems, volume 1402 of Lecture Notes in Math., pages 99-113. Springer Verlag, 1989.
     

    A. BACHELOT. Global solutions for nonlinear Dirac equations. In Integrables Systemes and applications, volume 342 of Lecture Notes in Physics, pages 1-11. Springer Verlag, 1989.
     

    A. BACHELOT. Opérateur de diffraction pour le système de Maxwell en métrique de Schwarzschild. In Equations aux dérivées partielles, Saint Jean de Monts, pages III.1-III.11, 1990.
     

    A. BACHELOT. Scattering by black-hole for electromagnetic field. In Inverse Methods in Action, pages 174-181. Springer Verlag, 1990.
     

    A. BACHELOT. Scattering operator for Maxwell equations outside Schwarzschild black-hole. In Integral equations and Inverse problems, volume 235 of Research Notes in Mathematics, pages 38-48. Pitman, 1991
     

    A. BACHELOT. Scattering for Maxwell equations on Schwarzschild metric. In "International Congress of Mathematicians", page 209. Kyoto, 1990.
     

    A. BACHELOT. Scattering of electromagnetic field by de Sitter-Schwarzschild black-hole. In Non linear hyperbolic equations and field theory, volume 253 of Research Notes in Mathematics, pages 23-35. Pitman, 1992.
     

    A. BACHELOT, A. PUJOLS. Boundary integral equation method in time domain for Maxwell's system. In 10th International Conference on Computing Methods in Applied Sciences and Engineering, pages 197-206. Nova Science Publishers, 1992.
     

    A. BACHELOT. Calcul des résonances d'un trou noir. In Ecole des ondes, pages 1-28. INRIA, 1993.
     

    A. BACHELOT. La diffraction en métrique de Schwarzschild : complétude asymptotique et résonances. In Séminaire X-EDP, pages VIII.1-VIII.13. Ecole Polytechnique, 1993.
     

    A. BACHELOT,  A. MOTET-BACHELOT. Scattering resonances for Schwarzschild black-hole. In Oliveri Donato, editor, Nonlinear Hyperbolic Problems : Theoretical, Applied, and Computational Aspects of Wave Propagation Phenomena, volume 43 of Vieweg Notes, pages 33-40, 1993.
     

    A. BACHELOT, V. LANGE. Time dependent integral method for Maxwell's system. In G. Cohen, editor, Third International Conference on Mathematical and Numerical Aspects of Wave Propagation Phenomena, pages151-159. SIAM-INRIA, 1995.
     

    A. BACHELOT, V. LANGE. Time dependent integral method for Maxwell's system with impedance boundary condition. In 10th International Conference on Boundary Element Technology, BETECH95, pages 137-144. Computational Mechanics Publications, 1995.
     

    A. BACHELOT, V. LUBET. On the coupling of boundary element and finite element methods for a time problem. In G. Cohen, editor, Third International Conference on Mathematical and Numerical Aspects of Wave Propagation Phenomena, pages 130-139. SIAM-INRIA, 1995.
     

    A. BACHELOT. Diffusion classique et quantique par un trou noir en formation. In Séminaire X-EDP, pages XV.1-XV.18. Ecole Polytechnique, 1996.
     

    A. BACHELOT, P. CHARRIER, A. PUJOLS, D. ROUART. Parallel Algorithm for Solving Time Convolution Equations and Application to CEM. In 8th SIAM Conference on Parallel Processing for Scientific Computing, Minneapolis 1997, (CD-ROM). SIAM, 1997.
     

    A. BACHELOT. Scattering operator for the Klein-Gordon equation on the Schwarzschild metric. In International Conference on Non Linear Evolution Partial Differential Equations, Beijing 93, AMS/IP Studies in Advanced Math., volume 3, 1997.
     

    A. BACHELOT. L'effet Hawking. in Séminaire X-EDP, Ecole Polytechnique 1997.
     

    A. BACHELOT, J-P. RAOULT. Mathématiques et entreprises. In Mathématiques A Venir, SMF-SMAI, Gazette des mathématiciens, supp. au n°75, 1997.
     

    A. BACHELOT, L. BOUNHOURE, A. PUJOLS. Coupling of finite elements and retarded potentials for an electromagnetic scattering problem by an inhomogeneous obstacle. In Fourth International Conference on Mathematical and Numerical Aspects of Wave Propagation Phenomena, Golden, Colorado, SIAM, 1998.
     

    A. BACHELOT. Wave equation and causality violation. in Séminaire X-EDP, Ecole Polytechnique 2001.
     

    A. BACHELOT, E. DARRIGRAND, K. MER-NKONGA. Coupling of a Fast Multipole Method and a Microlocal Discretization for Integral Equations of Electromagnetism. In  "WAVES 2003, Jyväskylä, Finlande, SIAM-INRIA, Springer Verlag, 2003.
     

    A. BACHELOT. Paradoxe de Klein et diffusion pour l'équation de Klein-Gordon chargée. in Séminaire X-EDP, Ecole Polytechnique 2004.
     

    L. GATARD, A. BACHELOT, K. MER-NKONGA. High order boundary integral methods for Maxwell²s equations: coupling of microlocal discretization and fast multipole methods. In "ENUMATH 05", Springer-Verlag, 2005.
     

    J. MORICE, K. MER-NKONGA, A. BACHELOT.  Fast multipole method for solving the radiosity equation. In "ENUMATH 05", Springer-Verlag, 2005.
     

    J. MORICE, K. MER-NKONGA, A. BACHELOT. Comparison of different fast multipole methods for solving the radiosity equation. In "Advances in Boundary Element Techniques VII", M.H. Aliabadi, B. Gatmiri, A. Sellier edts., 2006.
     

    J. MORICE, K. MER-NKONGA, A. BACHELOT. Numerical comparison of different fast multipole method for the radiosity equation. In "WAVES 07", Springer, 2007. 


    Publications d'intérêt général / Miscellanea 


    A. BACHELOT, C. VIDAL Edts. La recherche en sciences : Morceaux choisis. Publications de l'Université Bordeaux-1, 1994.

     

    A. BACHELOT. J-P. RAOULT. Mathématiques et entreprises. In "Mathématiques A Venir", SMF-SMAI, Gazette des mathématiciens, supp. au numéro 75, 1997.
     

    A. BACHELOT,  R. DIENG-KUNTZ, E. DUBOIS-VIOLETTE, L. HEMIDY, C. HERMANN, M. IMBERT, J. MASSOT, L. NURIT, F. THEBAUD. Rapport sur l'égalité professionnelle entre les femmes
    et les hommes dans l'enseignement supérieur et la recherche
    . Ministère délégué à l'enseignement supérieur et à la recherche, 2006, disponible sur

    http://www.ladocumentationfrancaise.fr/rapports-publics/064000887/index.shtml

    A. BACHELOT. Les mathématiques des trous noirs, in Brèves de maths, Martin Andler, Liliane Bel, Sylvie Benzoni, Thierry Goudon, Cyril Imbert, Antoine Rousseau, Editions Nouveau Monde, 2014, en ligne sur

    http://www.breves-de-maths.fr/les-mathematiques-des-trous-noirs/

    A. BACHELOT. Voir les trous noirs, in Brèves de maths, Martin Andler, Liliane Bel, Sylvie Benzoni, Thierry Goudon, Cyril Imbert, Antoine Rousseau, Editions Nouveau Monde, 2014, en ligne sur

    http://www.breves-de-maths.fr/voir-les-trous-noirs/


      VIDEOS


    Colloquium, Université de Nice, mars 2016 : "Questions de cosmologie mathématique"
    http://unspod.unice.fr/video/3871-questions-de-cosmologie-mathematique-alain-bachelot-colloquium-du-ljad/


    Retour à la page d'accueil. / Back to Home Page

  • A. BACHELOT. The Dirac System on the Anti-De Sitter Universe. Comm. Math. Phys., 2008.

    We investigate the global solutions of the Dirac equation on the Anti-de-Sitter Universe. Since this space is not globally hyperbolic, the Cauchy problem is not, a priori, well-posed. Nevertheless we can prove that there exists unitary dynamics, but its uniqueness crucially depends on the ratio beween the mass $M$ of the field and the cosmological constant $\Lambda > 0$ : it appears a critical value, $\Lambda /12$, which plays a role similar to the Breitenlohner-Freedman bound for the scalar fields. When $M^2 \geq \Lambda/12$ there exists a unique unitary dynamics. In opposite, for the light fermions satisfying $M^2 < \Lambda/12$, we construct several asymptotic conditions at infinity, such that the problem becomes well-posed. In all the cases, the spectrum of the hamiltonian is discrete. We also prove a result of equipartition of the energy.

  • A. BACHELOT. Wave Propagation and Scattering for the RS2 Brane Cosmology Model. J. Hyperbolic Differ. Equ., 2009.

    We study the wave equation for the gravitational fluctuations in the Randall-Sundrum brane cosmology model. We solve the global Cauchy problem and we establish that the solutions are the sum of a slowly decaying massless wave localized near the brane, and a superposition of massive dispersive waves. We compute the kernel of the truncated resolvent. We prove some $L^1-L^{\infty}$, $L^2-L^{\infty}$ decay estimates and global $L^p$ Strichartz type inequalities. We develop the complete scattering theory : existence and asymptotic completness of the wave operators, computation of the scattering matrix, determination of the resonances on the logarithmic Riemann surface.

  • A. BACHELOT. The Klein-Gordon Equation in Anti-de Sitter Cosmology, J. Math. Pures Appl., 96 : 527–554, 2011.

    This paper deals with the Klein-Gordon equation on the Poincar\'e chart of the 5-dimensional Anti-de Sitter universe. When the mass $\mu$ is larger than $-\frac{1}{4}$, the Cauchy problem is well posed despite the loss of global hyperbolicity due to the time-like horizon. We express the finite energy solutions in the form of a continuous Kaluza-Klein tower and we deduce a uniform decay as $\mid t\mid^{-\frac{3}{2}}$. We investigate the case $\mu=\frac{\nu^2-1}{2}$, $\nu\in\NN^*$, which encompasses the gravitational fluctuations, $\nu=4$, and the electromagnetic waves, $\nu=2$. The propagation of the wave front set shows that the horizon acts like a perfect mirror. We establish that the smooth solutions decay as  $\mid t\mid^{-2-\sqrt{\mu+\frac{1}{4}}}$, and we get global $L^p$ estimates of Strichartz type. When $\nu$ is even, there appears a lacuna and the equipartition of the energy occurs at finite time for the compactly supported initial data, although the Huygens principle fails. We address the cosmological model of the negative tension Minkowski brane, on which a Robin boundary condition is imposed. We prove the hyperbolic mixed problem is well-posed and the normalizable solutions can be expanded into a discrete Kaluza-Klein tower. We establish some $L^2-L^{\infty}$ estimates in suitable weighted Sobolev spaces.

  • A. BACHELOT. New Dynamics in the Anti-de Sitter Universe AdS^5, Comm. Math. Phys., 320, 723-759, 2013.

    This paper deals with the propagation of the gravitational waves in the Poincar ́e patch of the 5-dimensional Anti-de Sitter universe. We construct a large family of unitary dynamics with respect to some high order energies that are conserved and positive. These dynamics are associated with asymptotic conditions on the conformal time-like boundary of the universe. This result does not contradict the statement of Breitenlohner-Freedman that the hamiltonian is essentially self- adjoint in L2 and thus accordingly the dynamics is uniquely determined. The key point is the introduction of a new Hilbert functional framework that contains the massless graviton which is not normalizable in L2. Then the hamiltonian is not essentially self-adjoint in this new space and possesses a lot of different positive self-adjoint extensions. These dynamics satisfy a holographic principle : there exists a renormalized boundary value which completely characterizes the whole field in the bulk.

  • A. BACHELOT. On the Klein-Gordon equation near a De Sitter brane in an Anti-de Sitter bulk,  arXiv:1402.1071v3

    In this paper we investigate the Klein-Gordon equation in the past causal domain of a De Sitter brane imbedded in a Anti-de Sitter bulk. We solve the global mixed hyperbolic problem. We prove that any finite energy solution can be expressed as a Kaluza-Klein tower that is a superposition of free fields in the Steady State Universe, of which we study the asymptotic behaviours. We show that the leading term of a gravitational fluctuation is a massless graviton, i.e. the De Sitter brane is linearly stable.

  • A. BACHELOT. Waves in the Witten Bubble of Nothing and the Hawking Wormhole, arXiv:1601.03682
  • We investigate the propagation of the scalar waves in the Witten space-time called "bubble of nothing" and in its remarkable sub-manifold, the lorentzian Hawking wormhole. Due to the global hyperbolicity, the global Cauchy problem is well-posed in the fonctional framework associated with the energy. We perform a complete spectral analysis that allows to get an explicit form of the solutions in terms of special functions. When the effective mass is non zero, the profile of the waves is asymptotically almost periodic in time. In opposite, the massless case is dispersive. We develop the scattering theory, classical as well as quantum. There is no creation of particle. The resonances can be defined in the massless case and they are purely imaginary.

  • A. BACHELOT-MOTET, A. BACHELOT. Waves on accelerating dodecahedral universes, arXiv:1609.00806
  • We investigate the wave propagation on a compact 3-manifold of constant positive curvature with a non trivial topology, the Poincar\'e dodecahedral space, when the scale factor is exponentially increasing. We prove the existence of a limit state as t tends to infinity and we get its analytic expression. The deep sky is described by this asymptotic profile thanks to the Sachs-Wolfe formula. We transform the Cauchy problem into a mixed problem posed on a fundamental domain determined by the quaternionic calculus. We perform an accurate scheme of computation: we employ a variational method using a space of second order finite elements that is invariant under the action of the binary icosahedral group.

  • A. BACHELOT. Wave asymptotics at a cosmological time-singularity, arXiv:1806.01543
  • We investigate the propagation of the scalar waves in the FLRW universes beginning with a Big Bang and ending with a Big Crunch, a Big Rip, a Big Brake or a Sudden Singularity. We obtain the sharp description of the asymptotics for the solutions of the linear Klein-Gordon equation, and similar results for the  semilinear equation with a subcritical exponent. We prove that the number of cosmological particle creations is finite under general assumptions on the initial Big Bang and the final Big Crunch or Big Brake.


    Conference "Equations hyperboliques et physique mathématique"   on the Hyperspace (NEW)

    Preprints

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