Orbital stability of standing waves for some nonlinear Schrödinger equations

Abstract

We present a general method which enables us to prove the orbital stability of some standing waves in nonlinear Schrödinger equations. For example, we treat the cases of nonlinear Schrödinger equations arising in laser beams, of time-dependent Hartree equations ....

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Adams, R.A., Clarke, F.H.: Gross's logarithmic sobolev inequality: a simple proof (preprint)

  2. 2.

    Berestycki, H., Cazenave, T.: To appear

  3. 3.

    Berestycki, H., Lions, P.L.: Existence d'ondes solitaires dans des problèmes nonlinéaires du type Klein-Gordon. C. R. Paris287, 503–506 (1978);288, 395–398 (1979)

    Google Scholar 

  4. 4.

    Berestycki, H., Lions, P.L.: Nonlinear scalar fields equations. Parts I and II. Arch. Rat. Mech. Anal. (to appear)

  5. 5.

    Berger, M.S.: On the existence and structure of stationary states for a nonlinear Klein-Gordon equation. J. Funct. Anal.9, 249–261 (1972)

    Google Scholar 

  6. 6.

    Bialynicki-Birula, I., Mycielski, J.: Nonlinear wave mechanics. Ann. Phys.100, 62–93 (1976)

    Google Scholar 

  7. 7.

    Cazenave, T.: Equations de Schrödinger non linéaires. Thèse de 3ème cycle Univ. P. et M. Curie, Paris (1978)

  8. 8.

    Cazenave, T.: Equations de Schrödinger non linéaires en dimension deux. Proc. R. Soc. Edin88, 327–346 (1979)

    Google Scholar 

  9. 9.

    Cazenave, T.: Stable solutions of the logarithmic Schrödinger equation. Nonlinear Anal. T.M.A. (to appear)

  10. 10.

    Ginibre, J., Velo, G.: On a class of nonlinear Schrödinger equations. I. The Gauchy problem, general case. J. Funct. Anal.32, 1–32 (1979)

    Google Scholar 

  11. 11.

    Ginibre, J., Velo, G.: On a class of nonlinear Schrödinger equations. III. Special theories in dimensions 1, 2, and 3. Ann. Inst. Henri Poincaré28, 287–316 (1978)

    Google Scholar 

  12. 12.

    Ginibre, J., Velo, G.: Equation de Schrödinger non linéaire avec interaction non locale. C. R. Paris288, 683–685 (1979)

    Google Scholar 

  13. 13.

    Ginibre, J., Velo, G.: On a class of nonlinear Schrödinger equations with non local interaction. Math. Zeitschr. (to appear)

  14. 14.

    Glassey, R.T.: On the blowing-up of solutions to the Cauchy Problem for nonlinear Schrödinger equations. J. Math. Phys.18, 1794–1797 (1977)

    Google Scholar 

  15. 15.

    Hartree, D.: The wave mechanics of an atom with a non-Coulomb central field. Part I. Theory and methods. Proc. Camb. Philos. Soc.24, 89–132 (1968)

    Google Scholar 

  16. 16.

    Kelley, P.L.: Self-focusing of optical beams. Phys. Rev. Lett.15, 1005–1008 (1965)

    Google Scholar 

  17. 17.

    Lieb, E.H.: Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation. Stud. Appl. Math.57, 93–105 (1977)

    Google Scholar 

  18. 18.

    Lieb, E.H., Simon, B.: The Hartree-Fock theory for Coulomb systems. Commun. Math. Phys.53, 185–194 (1974)

    Google Scholar 

  19. 19.

    Lions, P.L.: The Choquard equation and related equations. Nonlinear Anal. T.M.A.4, 1063–1073 (1980)

    Google Scholar 

  20. 20.

    Lions, P.L.: Some remarks on Hartree equation. Nonlinear Anal. T.M.A.5, 1245–1256 (1981)

    Google Scholar 

  21. 21.

    Lions, P.L.: Principe de concentration — compacité en calcul des variations. C. R. Paris294, 261–264 (1982)

    Google Scholar 

  22. 22.

    Lions, P.L.: To appear

  23. 23.

    Lin, J.E., Strauss, W.: Decay and scattering of solutions of a nonlinear Schrödinger equation. J. Funct. Anal.30, 245–263 (1978)

    Google Scholar 

  24. 24.

    MacLeod, K., Serrin, J.: Personal communication

  25. 25.

    Nehari, Z.: On a nonlinear differential equation arising in nuclear physics. Proc. R. Irish Acad.62, 117–135 (1963)

    Google Scholar 

  26. 26.

    Pecher, H., Von Wahl, W.: Time dependent nonlinear Schrödinger equations. Manuscripta Mathematica (to appear)

  27. 27.

    Reeken, M.: Global theorem on bifurcation and its application to the Hartree equation of the Helium atom. J. Math. Phys.11, 2505–2512 (1970)

    Google Scholar 

  28. 28.

    Ryder, G.: Boundary value problems for a class of nonlinear differential equations. Pac. J. Math.22, 477–503 (1967)

    Google Scholar 

  29. 29.

    Slater, J.C.: A note on Hartree's method. Phys. Rev.35, 210–211 (1930)

    Google Scholar 

  30. 30.

    Strauss, W.: Existence of solitary waves in higher dimensions. Commun. Math. Phys.55, 149–162 (1977)

    Google Scholar 

  31. 31.

    Strauss, W.: The nonlinear Schrödinger equation. Proceedings of the Rio conference, August 1977

  32. 32.

    Stuart, C.A.: Existence theory for the Hartree equation. Arch. Rat. Mech. Anal.51, 60–69 (1973)

    Google Scholar 

  33. 33.

    Stuart, C.A.: An example in nonlinear functional analysis: the Hartree equation. J. Math. Anal. Appl.49, 725–733 (1975)

    Google Scholar 

  34. 34.

    Suydam, B.R.: Self-focusing of very powerful laser beams. U.S. Dept. of Commerce. N.B.S. Special Publication 387

Download references

Author information

Affiliations

Authors

Additional information

Communicated by B. Simon

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Cazenave, T., Lions, P.L. Orbital stability of standing waves for some nonlinear Schrödinger equations. Commun.Math. Phys. 85, 549–561 (1982). https://doi.org/10.1007/BF01403504

Download citation

Keywords

  • Neural Network
  • Statistical Physic
  • Laser Beam
  • Complex System
  • Nonlinear Dynamics