Archive for Rational Mechanics and Analysis

, Volume 136, Issue 3, pp 291–303 | Cite as

Global solvability of the maxwell-bloch equations from nonlinear optics

  • P. Donnat
  • J. Rauch


Neural Network Complex System Nonlinear Dynamics Electromagnetism Nonlinear Optic 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [BL]
    D. M. Bishop &B. Lam, Ab initio study of third order nonlinear optical properties of helium,Phys. Rev. A,37, 464–469 (1988).Google Scholar
  2. [BC]
    P. Butcher &D. Cotter, The Elements of Nonlinear Optics, Cambridge Univ. Press, Cambridge, UK, 1990.Google Scholar
  3. [CGT]
    R. Y. Chiao, E. Garmire &C. H. Townes, Self-trapping of optical beams,Phys. Rev. Letters,13, 479–480 (1964).Google Scholar
  4. [DM]
    E. L. Dawes &J. H. Marburger, Computer studies in self-focusing,Phys. Rev.,179, 862–867 (1969).Google Scholar
  5. [DLP]
    A. P. Dyshko, V. N. Lugovoi &A. M. Prokhorov, Development of an optical waveguide in the propagation of light in nonlinear mediumSov. Phys. JETP,34, 1235–1241 (1972).Google Scholar
  6. [D]
    P. Donnat, Quelque contributions mathématiques en optique nonlinéaire, Thèse, Ecole Polytechnique, Paris, 1994.Google Scholar
  7. [DR]
    P. Donnat &J. Rauch, Dispersive nonlinear geometric optics,J. Math. Phys. to appear.Google Scholar
  8. [FF]
    M. D. Feit &J. A. Fleck, Beam non paraxiality, filament formation and beam break up in the self-focusing of optical beam,J. Opt. Soc. Am. B,5, 633–640 (1988).Google Scholar
  9. [G]
    R. T. Glassey, On the blow-up of solutions of the Cauchy problem for the nonlinear Schrödinger equation,J. Math. Physics,18, 1794–1797 (1977).Google Scholar
  10. [JMR]
    J.-L. Joly, G. Metivier &J. Rauch, Global solvability of the anharmonic oscillator model from nonlinear optics,SIAM J. Math. Anal.,27, 905–913 (1996).Google Scholar
  11. [H]
    L. Hörmander, Non-linear Hyperbolic Differential Equations, Lecture Notes, Lund University, 1987.Google Scholar
  12. [K]
    P. L. Kelley, Self-focussing of optical beams,Phys. Rev. Letters,15, 1005–1008 (1965).Google Scholar
  13. [L]
    G. G. Luther, A. C. Newell &J. V. Maloney, Normal dispersion arrests critical collapse, preprint.Google Scholar
  14. [MP]
    V. Malkin &G. Papanicolaou, On self-focusing of short laser pulses, preprint (1993).Google Scholar
  15. [M]
    J. H. Marburger, Self-focussing theory,Prog. Quant. Elect. 4, 35–110 (1975).Google Scholar
  16. [NM]
    A. Newell &J. Moloney, Nonlinear Optics, Addison-Wesley, Reading, Mass. 1992.Google Scholar
  17. [PP]
    R. Pantell &H. Puthoff, Fundamentals of Quantum Electronics, Wiley, New York, 1969.Google Scholar
  18. [Ra]
    J. Rauch, Partial Differential Equations, Springer-Verlag, New York, 1991.Google Scholar
  19. [Re]
    M. Reed, Abstract Non-Linear Wave Equations, Springer-Verlag, New York, 1975.Google Scholar
  20. [Rot1]
    J. E. Rothenberg, Pulse splitting during self-focussing in normally dispersive media.Opt. Letters,17, 583–585 (1992).Google Scholar
  21. [Rot2]
    J. E. Rothenberg, Breakdown of the slowly varying envelope approximation in the self-focussing of femtosecond pulses.Opt. Letters,17, 1340–1342 (1992).Google Scholar
  22. [Ro]
    C. Rouyer, E. Mazataud, I. Allais, A. Pierre, S. Seznec, C. Sauteret, G. Mourou &A. Migus: Generation of 50TW femtosecond pulses in a Ti: sapphire/Nd: glass chain,Opt. Letters,18, 214–216 (1993).Google Scholar
  23. [Sh]
    Y. R. Shen, The Principles of Nonlinear Optics, Wiley-Interscience, New York, 1984.Google Scholar
  24. [T]
    M. Taylor, Pseudodifferential Operators and Nonlinear Optics, Birkhäuser, Boston, 1991.Google Scholar
  25. [Tal]
    V. I. Talanov, Self-focusing of wave beams in nonlinear media,J. Exper. Theor. Phys. Lett.,2, 138–141 (1965).Google Scholar
  26. [Y]
    V. Yudovich, Nonstationary flows of an ideal incompressible fluid,Zh. Vych. Math.,3, 1032–1066 (1963).Google Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • P. Donnat
    • 1
    • 2
  • J. Rauch
    • 1
    • 2
  1. 1.Commisariat à l'Energie AtomiqueCentre d'Etude de Limeil ValentonVilleneuve St. Georges cedexFrance
  2. 2.Department of MathematicsUniversity of MichiganAnn Arbor

Personalised recommendations