Communications in Mathematical Physics

, Volume 282, Issue 3, pp 721–731 | Cite as

Multiple Bound States of Nonlinear Schrödinger Systems

Article

Abstract

This paper is concerned with existence of bound states for Schrödinger systems which have appeared as several models from mathematical physics. We establish multiplicity results of bound states for both small and large interactions. This is done by different approaches depending upon the sizes of the interaction parameters in the systems. For small interactions we give a new approach to deal with multiple bound states. The novelty of our approach lies in establishing a certain type of invariant sets of the associated gradient flows. For large interactions we use a minimax procedure to distinguish solutions by analyzing their Morse indices.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Akhmediev N., Ankiewicz A.: Partially coherent solitons on a finite background. Phys. Rev. Lett. 82, 2661–2664 (1999)CrossRefADSGoogle Scholar
  2. 2.
    Ambrosetti A., Colorado E.: Bound and ground states of coupled nonlinear Schrödinger equations. C. R. Math. Acad. Sci. Paris 342, 453–458 (2006)MATHMathSciNetGoogle Scholar
  3. 3.
    Ambrosetti A., Colorado E.: Standing waves of some coupled nonlinear Schrödinger equations. J. London Math. Soc. 75, 67–82 (2007)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Ambrosetti A., Rabinowitz P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bahri A., Lions P.L.: Solutions of superlinear elliptic equations and their Morse indices. Comm. Pure Appl. Math. 45, 1205–1215 (1992)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bartsch T., Pankov A., Wang Z.-Q.: Nonlinear Schrödinger equations with steep potential well. Comm. Contem. Math. 3, 1–21 (2001)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Bartsch T., Wang Z.-Q.: Note on ground states of nonlinear Schrödinger systems. J. Partial Differ. Eqs. 19, 200–207 (2006)MATHMathSciNetGoogle Scholar
  8. 8.
    Bartsch T., Wang Z.-Q., Wei J.C.: states for a coupled Schrödinger system. J. Fixed Point Th. Appl. 2, 353–367 (2007)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Chang K.C.: Infinite Dimensional Morse Theory and Multiple Solution Problems, Progress in Nonlinear Differential Equations and Their Applications, 6. Birkhäuser, Boston (1993)Google Scholar
  10. 10.
    Christodoulides D.N., Coskun T.H., Mitchell M., Segev M.: Theory of incoherent self-focusing in biased photorefractive media. Phys. Rev. Lett. 78, 646–649 (1997)CrossRefADSGoogle Scholar
  11. 11.
    Dancer, E.N., Wei, J.C., Weth, T.: A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system. PreprintGoogle Scholar
  12. 12.
    Deimling K.: Ordinary Differential Equations in Banach Spaces, Lecture Notes in Mathematics, vol. 596. Springer-Verlag, Berlin (1977)Google Scholar
  13. 13.
    Esry B.D., Greene C.H., Burke J.P. Jr., Bohn J.L.: Hartree-Fock theory for double condensates. Phys. Rev. Lett. 78, 3594–3597 (1997)CrossRefADSGoogle Scholar
  14. 14.
    Genkin, G.M.: Modification of superfluidity in a resonantly strongly driven Bose-Einstein condensate. Phys. Rev. A, 65 (2002), No.035604Google Scholar
  15. 15.
    Hioe F.T.: Solitary waves for N coupled nonlinear Schrödinger equations. Phys. Rev. Lett. 82, 1152–1155 (1999)CrossRefADSGoogle Scholar
  16. 16.
    Hioe F.T., Salter T.S.: Special set and solutions of coupled nonlinear Schrödinger equations. J. Phys. A: Math. Gen. 35, 8913–8928 (2002)MATHCrossRefADSMathSciNetGoogle Scholar
  17. 17.
    Kanna, T., Lakshmanan, M.: Exact soliton solutions, shape changing collisions, and partially coherent solitons in coupled nonlinear Schrödinger equations. Phys. Rev. Lett., 86, 5043(1–4) (2001)Google Scholar
  18. 18.
    Lin T.-C., Wei J.C.: Ground state of N coupled nonlinear Schrödinger equations in R n, n ≤  3. Commun. Math. Phys. 255, 629–653 (2005)MATHCrossRefADSMathSciNetGoogle Scholar
  19. 19.
    Lin T.-C., Wei J.C.: Spikes in two coupled nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 22, 403–439 (2005)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Lin T.-C., Wei J.C.: Spikes in two-component systems of nonlinear Schrödinger equations with trapping potentials. J. Differ. Eq. 299(2), 538–569 (2006)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Maia L.A., Montefusco E., Pellacci B.: Positive solutions for a weakly coupled nonlinear Schrödinger system. J. Differ. Eq. 229, 743–767 (2006)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Montefusco, E., Pellacci, B., Squassina, M.: Semiclassical states for weakly coupled nonlinear Schrödinger systems. PreprintGoogle Scholar
  23. 23.
    Mitchell M., Chen Z., Shih M., Segev M.: Self-Trapping of partially spatially incoherent light. Phys. Rev. Lett. 77, 490–493 (1996)CrossRefADSGoogle Scholar
  24. 24.
    Pomponio A.: Coupled nonlinear Schrödinger systems with potentials. J. Differ. Eq. 227, 258–281 (2006)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Rabinowitz, P.H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Reg. Conf. Ser. Math. 65, Providence, R.I.: Amer. Math. Soc., 1986Google Scholar
  26. 26.
    Sirakov B.: Least energy solitary waves for a system of nonlinear Schrödinger equations in \({\mathbb{R}^n}\). Commun. Math. Phys. 271, 199–221 (2007)MATHCrossRefADSMathSciNetGoogle Scholar
  27. 27.
    Timmermans E.: Phase separation of Bose-Einstein condensates. Phys. Rev. Lett. 81, 5718–5721 (1998)CrossRefADSGoogle Scholar
  28. 28.
    Wei J.C., Weth T.: Nonradial symmetric bound states for a system of two coupled Schrödinger equations. Rend. Lincei Mat. Appl. 18, 279–294 (2007)MathSciNetGoogle Scholar
  29. 29.
    Wei, J.C., Weth, T.: Radial solutions and phase seperation in a system of two coupled Schrödinger equations. Arch. Rational Mech. Anal., On-line First, doi:10:1007/s00205-008-0121-9, 2008

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.School of Mathematical SciencesCapital Normal UniversityBeijingP.R. China
  2. 2.Department of Mathematics and StatisticsUtah State UniversityLoganUSA

Personalised recommendations