Skip to main content
SpringerLink
Log in

Existence and Stability of Standing Waves for Supercritical NLS with a Partial Confinement

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We prove the existence of orbitally stable ground states to NLS with a partial confinement together with qualitative and symmetry properties. This result is obtained for nonlinearities which are L 2-supercritical; in particular, we cover the physically relevant cubic case. The equation that we consider is the limit case of the cigar-shaped model in BEC.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Anderson M.H., Ensher J.R., Matthews M.R., Wieman C.E., Cornell E.A.: Bose–Einstein condensation in a dilute atomic vapor. Science 269, 14 (1995)

    Article  Google Scholar 

  2. Antonelli P., Carles R., Drumond Silva J.: Scattering for nonlinear Schrödinger equation under partial harmonic confinement. Commun. Math. Phys. 334(1), 367–396 (2015)

    Article  ADS  MATH  Google Scholar 

  3. Bao W., Cai Y.: Mathematical theory and numerical methods for Bose–Einstein condensation. Kinet. Relat. Models 6(1), 1–135 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bao W., Jaksch D., Markowich P.A.: Numerical solution of the Gross–Pitaevskii equation for Bose–Einstein condensation. J. Comput. Phys. 187(1), 318–342 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Bellazzini J., Jeanjean L.: On dipolar quantum gases in the unstable regime. SIAM J. Math. Anal. 48(3), 2028–2058 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bellazzini J., Jeanjean L., Luo T.: Existence and instability of standing waves with prescribed norm for a class of Schrödinger–Poisson equations. Proc. Lond. Math. Soc. (3) 107(2), 303–339 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Benci V., Visciglia N.: Solitary waves with non-vanishing angular momentum. Adv. Nonlinear Stud. 3(1), 151–160 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Berestycki H., Cazenave T.: Instabilité des états stationnaires dans les équations de Schrödinger et de Klein–Gordon non linéaires. C. R. Acad. Sci. Paris Sér. I Math. 293(9), 489–492 (1981)

    MathSciNet  MATH  Google Scholar 

  9. Brézis H., Lieb E.: A relation between pointwise convergence of functions and convergence of functionals. Proc. Am. Math. Soc. 88(3), 486–490 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  10. Brock F., Solynin A.Y.: An approach to symmetrization via polarization. Trans. Am. Math. Soc. 352(4), 1759–1796 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  11. Brothers J.E., Ziemer W.P.: Minimal rearrangements of Sobolev functions. J. Reine Angew. Math. 384, 153–179 (1988)

    MathSciNet  MATH  Google Scholar 

  12. Byeon J., Jeanjean L., Mariş M.: Symmetry and monotonicity of least energy solutions. Calc. Var. Partial Differ. Equ. 36(4), 481–492 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Carles R.: Critical nonlinear Schrödinger equations with and without harmonic potential. Math. Models Methods Appl. Sci. 12(10), 1513–1523 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cazenave, T.: Semilinear Schrödinger equations, vol. 10 of Courant Lecture Notes in Mathematics. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI (2003)

  15. Cazenave T., Lions P.-L.: Orbital stability of standing waves for some nonlinear Schrödinger equations. Commun. Math. Phys. 85(4), 549–561 (1982)

    Article  ADS  MATH  Google Scholar 

  16. Dalfovo F., Giorgini S., Pitaevskii L.P., Stringari S.: Theory of Bose–Einstein condensation in trapped gases. Rev. Mod. Phys. 71(3), 463 (1999)

    Article  ADS  Google Scholar 

  17. Erdős L., Schlein B., Yau H.-T.: Derivation of the Gross–Pitaevskii hierarchy for the dynamics of Bose–Einstein condensate. Commun. Pure Appl. Math. 59(12), 1659–1741 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Erdős L., Schlein B., Yau H.-T.: Rigorous derivation of the Gross–Pitaevskii equation with a large interaction potential. J. Am. Math. Soc. 22(4), 1099–1156 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Fukuizumi R.: Stability and instability of standing waves for the nonlinear Schrödinger equation with harmonic potential. Discrete Contin. Dyn. Syst. 7(3), 525–544 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  20. Fukuizumi R., Hadj Selem F., Kikuchi H.: Stationary problem related to the nonlinear Schrödinger equation on the unit ball. Nonlinearity 25(8), 2271–2301 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. Fukuizumi R., Ohta M.: Instability of standing waves for nonlinear Schrödinger equations with potentials. Differ. Integral Equ. 16(6), 691–706 (2003)

    MATH  Google Scholar 

  22. Fukuizumi R., Ohta M.: Stability of standing waves for nonlinear Schrödinger equations with potentials. Differ. Integral Equ. 16(1), 111–128 (2003)

    MATH  Google Scholar 

  23. Griffin A., Snoke D., Stringari S.: Bose-Einstein Condensation. Cambridge University Press, Cambridge (1996)

    Google Scholar 

  24. Hajaiej H., Stuart C.A.: Symmetrization inequalities for composition operators of Carathéodory type. Proc. Lond. Math. Soc. (3) 87(2), 396–418 (2003)

    Article  MATH  Google Scholar 

  25. Hajaiej H., Stuart C.A.: On the variational approach to the stability of standing waves for the nonlinear Schrödinger equation. Adv. Nonlinear Stud. 4(4), 469–501 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  26. Jeanjean L.: Existence of solutions with prescribed norm for semilinear elliptic equations. Nonlinear Anal. 28(10), 1633–1659 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kavian O.: Introduction à la Théorie des Points Critiques et Applications aux Problèmes Elliptiques, vol. 13 of Mathématiques and Applications (Berlin) [Mathematics and Applications]. Springer, Paris (1993)

    MATH  Google Scholar 

  28. Kenig, C.E.: Carleman estimates, uniform Sobolev inequalities for second-order differential operators, and unique continuation theorems. In: Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), pp. 948–960. Am. Math. Soc., Providence, RI (1987)

  29. Li Y., Ni W.-M.: Radial symmetry of positive solutions of nonlinear elliptic equations in R n. Commun. Partial Differ. Equ. 18(5–6), 1043–1054 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  30. Lieb E., Loss M.: Analysis, Volume 14 of Graduate Studies in Mathematics, 2nd edn. American Mathematical Society, Providence (2001)

    Google Scholar 

  31. Lieb E., Seiringer R.: Proof of Bose–Einstein condensation for dilute trapped gases. Phys. Rev. Lett. 88(17), 170409 (2002)

    Article  ADS  Google Scholar 

  32. Lieb E., Seiringer R., Solovej J.P., Yngvason J.: The Mathematics of the Bose Gas and Its Condensation, Volume 34 of Oberwolfach Seminars. Birkhäuser Verlag, Basel (2005)

    MATH  Google Scholar 

  33. Lopes O.: Radial symmetry of minimizers for some translation and rotation invariant functionals. J. Differ. Equ. 124(2), 378–388 (1996)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  34. Noris B., Tavares H., Verzini G.: Existence and orbital stability of the ground states with prescribed mass for the L 2-critical and supercritical NLS on bounded domains. Anal. PDE 7(8), 1807–1838 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  35. Rousset F., Tzvetkov N.: Stability and instability of the KdV solitary wave under the KP-I flow. Commun. Math. Phys. 313(1), 155–173 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  36. Stuart, C.A.: Private Communication. (2016)

  37. Terracini S., Tzvetkov N., Visciglia N.: The nonlinear Schrödinger equation ground states on product spaces. Anal. PDE 7(1), 73–96 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  38. Weinstein M.I.: Lyapunov stability of ground states of nonlinear dispersive evolution equations. Commun. Pure Appl. Math. 39(1), 51–67 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  39. Willem M.: Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston (1996)

    Google Scholar 

  40. Zhang J.: Stability of standing waves for nonlinear Schrödinger equations with unbounded potentials. Z. Angew. Math. Phys. 51(3), 498–503 (2000)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Università di Sassari, Via Piandanna 4, 07100, Sassari, Italy

    Jacopo Bellazzini

  2. Laboratoire de Mathématiques (UMR 6623), University Bourgogne Franche-Comté, 16, Route de Gray, 25030, Besançon Cedex, France

    Nabile Boussaïd & Louis Jeanjean

  3. Dipartimento di Matematica, Università Degli Studi di Pisa, Largo Bruno Pontecorvo 5, 56127, Pisa, Italy

    Nicola Visciglia

Authors
  1. Jacopo Bellazzini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nabile Boussaïd
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Louis Jeanjean
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Nicola Visciglia
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nabile Boussaïd.

Additional information

Communicated by W. Schlag

This work has been carried out in the framework of the Project NONLOCAL (ANR-14-CE25-0013), funded by the French National Research Agency (ANR). J.B. was supported by Gnampa project 2016 “Equazioni nonlineari dispersive”. N.V. was supported by the research project PRA 2016.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bellazzini, J., Boussaïd, N., Jeanjean, L. et al. Existence and Stability of Standing Waves for Supercritical NLS with a Partial Confinement. Commun. Math. Phys. 353, 229–251 (2017). https://doi.org/10.1007/s00220-017-2866-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-017-2866-1

Search

Navigation