Advertisement

Archive for Rational Mechanics and Analysis

, Volume 194, Issue 3, pp 717–741 | Cite as

Multipulse Phases in k-Mixtures of Bose–Einstein Condensates

  • Susanna TerraciniEmail author
  • Gianmaria Verzini
Article

Abstract

For the system
$$-\Delta U_i+ U_i=U_i^3-\beta U_i\sum_{j\neq i}U_j^2,\quad i=1,\dots,k,$$
(with k ≧ 3), we prove the existence for β large of positive radial solutions on \({\mathbb R^N}\) . We show that as β →  + ∞, the profile of each component U i separates, in many pulses, from the others. Moreover, we can prescribe the location of such pulses in terms of the oscillations of the changing-sign solutions of the scalar equation  − ΔW  +  W  =  W3. Within an Hartree–Fock approximation, this provides a theoretical indication of phase separation into many nodal domains for the k-mixtures of Bose–Einstein condensates.

Keywords

Solitary Wave Implicit Function Theorem Einstein Condensate Radial Solution Nodal Domain 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ambrosetti, A., Colorado, E.: Standing waves of some coupled nonlinear Schrödinger equations. J. Lond. Math. Soc. 75(2), 67–82 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bartsch T., Willem M. (1993) Infinitely many radial solutions of a semilinear elliptic problem on RN. Arch. Rational Mech. Anal. 124: 261–276ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chang S.M., Lin C.S., Lin T.C., Lin W.W. (2004) Segregated nodal domains of two-dimensional multispecies Bose-Einstein condensates. Phys. D. 196: 341–361MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Conti M., Terracini S., Verzini G. (2002) Nehari’s problem and competing species systems. Ann. Inst. H. Poincaré Anal. Non Linéaire 19: 871–888ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Conti M., Terracini S., Verzini G. (2003) An optimal partition problem related to nonlinear eigenvalues. J. Funct. Anal. 198: 160–196MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dancer E.N., Wei J.C., Weth T. (2007) A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system. preprintGoogle Scholar
  7. 7.
    Jones C., Küpper T. (1986) On the infinitely many solutions of a semilinear elliptic equation. SIAM J. Math. Anal. 17: 803–835MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Lin T.C., Wei J.C. (2005) Ground state of N coupled nonlinear Schrödinger equations in Rn, n≦  3. Comm. Math. Phys. 255: 629–653ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Lin T.C., Wei J.C. (2006) Spikes in two-component systems of nonlinear Schrödinger equations with trapping potentials. J. Differ. Equ. 229: 538–569ADSCrossRefzbMATHGoogle Scholar
  10. 10.
    Maia L.A., Montefusco E., Pellacci B.: Infinitely many radial solutions for a weakly coupled nonlinear Schrödinger system (preprint)Google Scholar
  11. 11.
    Miranda C. (1940) Un’osservazione su un teorema di Brouwer. Boll. Un. Mat. Ital. 3(2): 5–7MathSciNetzbMATHGoogle Scholar
  12. 12.
    Nehari Z. (1961) Characteristic values associated with a class of nonlinear second order differential equations. Acta Math. 105: 141–175MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Rüegg Ch. et al (2003) Bose–Einstein condensation of the triple states in the magnetic insulator tlcucl3. Nature 423: 62–65ADSCrossRefGoogle Scholar
  14. 14.
    Sirakov B. (2007) Least energy solitary waves for a system of nonlinear Schrödinger equations in \({\mathbb{R}^n}\) . Comm. Math. Phys. 271: 199–221ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Strauss W.A. (1977) Existence of solitary waves in higher dimensions. Comm. Math. Phys. 55: 149–162ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Wei J.C., Weth T. Radial solutions and phase separation in a system of two coupled Schrödinger equations. Arch. Rational Mech. Anal. (to appear)Google Scholar
  17. 17.
    Wei J.C., Weth T. (2007) Nonradial symmetric bound states for a system of coupled Schrödinger equations. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl. 18(9): 279–293CrossRefzbMATHGoogle Scholar
  18. 18.
    Wei J.C., Weth T. (2008) Asymptotic behavior of solutions of planar systems with strong competition. Nonlinearity 21: 305–317ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Dipartimento di Matematica e ApplicazioniUniversità degli Studi di Milano-BicoccaMilanItaly
  2. 2.Dipartimento di MatematicaPolitecnico di MilanoMilanItaly

Personalised recommendations