Advertisement

Communications in Mathematical Physics

, Volume 318, Issue 1, pp 247–289 | Cite as

Stability and Symmetry-Breaking Bifurcation for the Ground States of a NLS with a δ′ Interaction

  • Riccardo AdamiEmail author
  • Diego Noja
Article

Abstract

We determine and study the ground states of a focusing Schrödinger equation in dimension one with a power nonlinearity |ψ|2μ ψ and a strong inhomogeneity represented by a singular point perturbation, the so-called (attractive) δ′ interaction, located at the origin. The time-dependent problem turns out to be globally well posed in the subcritical regime, and locally well posed in the supercritical and critical regime in the appropriate energy space. The set of the (nonlinear) ground states is completely determined. For any value of the nonlinearity power, it exhibits a symmetry breaking bifurcation structure as a function of the frequency (i.e., the nonlinear eigenvalue) ω. More precisely, there exists a critical value ω* of the nonlinear eigenvalue ω, such that: if ω0 <  ω <  ω*, then there is a single ground state and it is an odd function; if ω >  ω* then there exist two non-symmetric ground states. We prove that before bifurcation (i.e., for ω <  ω*) and for any subcritical power, every ground state is orbitally stable. After bifurcation (ω = ω* + 0), ground states are stable if μ does not exceed a value \({\mu^\star}\) that lies between 2 and 2.5, and become unstable for μ > μ*. Finally, for μ >  2 and \({\omega \gg \omega^*}\), all ground states are unstable. The branch of odd ground states for ω <  ω* can be continued at any ω >  ω*, obtaining a family of orbitally unstable stationary states. Existence of ground states is proved by variational techniques, and the stability properties of stationary states are investigated by means of the Grillakis-Shatah-Strauss framework, where some non-standard techniques have to be used to establish the needed properties of linearization operators.

Keywords

Solitary Wave Standing Wave Essential Spectrum Point Interaction Energy Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adami R., Noja D.: Existence of dynamics for a 1-d NLS equation in dimension one. J. Phys. A 42(49), 495302 (2009)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Adami, R., Noja, D., Visciglia, N.: Constrained energy minimization and ground states for NLS with point defects (2012, submitted), arXiv:1204.6344Google Scholar
  3. 3.
    Adami R., Cacciapuoti C., Finco D., Noja D.: On the structure of critical energy levels for the cubic focusing NLS on star graphs. J. Phys A 45, 192001 (2012)MathSciNetCrossRefADSGoogle Scholar
  4. 4.
    Adami R., Cacciapuoti C., Finco D., Noja D.: Stationary states of NLS on star graphs. Europhys. Lett. 100, 10003 (2012)CrossRefGoogle Scholar
  5. 5.
    Akhmediev N.N.: Novel class of nonlinear surface waves: asymmetric modes in a symmetric layered structure. Sov. Phys. JETP 56, 299–303 (1982)Google Scholar
  6. 6.
    Albeverio S., Brzeźniak Z., Dabrowski L.: Fundamental solutions of the Heat and Schrödinger Equations with point interaction. J. Func. An. 130, 220–254 (1995)zbMATHCrossRefGoogle Scholar
  7. 7.
    Albeverio, S., Gesztesy, F., Hoegh-Krohn, R., Holden, H.: Solvable Models in Quantum Mechanics. New York: Springer-Verlag, 1988Google Scholar
  8. 8.
    Avron J.E., Exner P., Last Y.: Periodic Schrödinger operators with large gaps and Wannier-Stark ladders. Phys. Rev. Lett. 72, 896–899 (1994)zbMATHCrossRefADSGoogle Scholar
  9. 9.
    Brezis H., Lieb E.H.: A relation between pointwise convergence of functions and convergence of functionals. Proc. Amer. Math. Soc. 88, 486–490 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Cazenave T., Lions P.-L.: Orbital stability of standing waves for some nonlinear Schrödinger equations. Commun. Math. Phys. 85, 549–561 (1982)MathSciNetzbMATHCrossRefADSGoogle Scholar
  11. 11.
    Cao X.D., Malomed A.B.: Soliton defect collisions in the nonlinear Schrödinger equation. Phys. Lett. A 206, 177–182 (1995)MathSciNetzbMATHCrossRefADSGoogle Scholar
  12. 12.
    Cazenave, T.: Semilinear Schrödinger Equations. Vol. 10, Courant Lecture Notes in Mathematics, Providence, RI: Amer. Math. Soc., 2003Google Scholar
  13. 13.
    Cheon T., Shigehara T.: Realizing discontinuous wave functions with renormalized short-range potentials. Phys. Lett. A 243, 111–116 (1998)CrossRefADSGoogle Scholar
  14. 14.
    Comech A., Pelinovsky D.: Purely nonlinear instability of standing waves with minimal energy. Comm. Pure App. Math. 56, 1565–1607 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Exner, P., Grosse, P.: Some properties of the one-dimensional generalized point interactions (a torso). http://arxiv.org/abs/math-ph/9910029v1, 1999
  16. 16.
    Exner P., Neidhart H., Zagrebnov V.A.: Potential approximations to δ′: an inverse Klauder phenomenon with norm-resolvent convergence. Commun. Math. Phys. 224, 593–612 (2001)CrossRefADSGoogle Scholar
  17. 17.
    Fibich G., Wang X.P.: Stability for solitary waves for nonlinear Schrödinger equations with inhomogenous nonlinearities. Physica D 175, 96–108 (2003)MathSciNetzbMATHCrossRefADSGoogle Scholar
  18. 18.
    Fukuizumi R., Jeanjean L.: Stability of standing waves for a nonlinear Schrödinger equation with a repulsive Dirac delta potential. Dis. Cont. Dyn. Syst. (A) 21, 129–144 (2008)MathSciNetGoogle Scholar
  19. 19.
    Fukuizumi R., Ohta M., Ozawa T.: Nonlinear Schrödinger equation with a point defect. Ann. Inst. H. Poincaré - AN 25, 837–845 (2008)MathSciNetzbMATHCrossRefADSGoogle Scholar
  20. 20.
    Fukuizumi R., Sacchetti A.: Bifurcation and stability for nonlinear Schrödinger equation with double well potential in the semiclassical limit. J. Stat. Phys. 145(6), 1546–1594 (2011)MathSciNetzbMATHCrossRefADSGoogle Scholar
  21. 21.
    Goodman R.H., Holmes P.J., Weinstein M.I.: Strong NLS soliton-defect interactions. Physica D 192, 215–248 (2004)MathSciNetzbMATHCrossRefADSGoogle Scholar
  22. 22.
    Grillakis M., Shatah J., Strauss W.: Stability theory of solitary waves in the presence of symmetry - I. J. Func. An. 74, 160–197 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Grillakis M., Shatah J., Strauss W.: Stability theory of solitary waves in the presence of symmetry - II. J. Func. An. 94, 308–348 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Hislop, P.D., Sigal, I.M.: Introduction to spectral theory: With applications to Schrödinger operators. New York: Springer, 1996Google Scholar
  25. 25.
    Haroske, D.D., Triebel, H.: Distributions, Sobolev Spaces, Elliptic Equations. Zürich: European Mathematical Society, 2008Google Scholar
  26. 26.
    Holmer J., Marzuola J., Zworski M.: Fast soliton scattering by delta impurities. Commun. Math. Phys 274, 187–216 (2007)MathSciNetzbMATHCrossRefADSGoogle Scholar
  27. 27.
    Jackson R.K., Weinstein M.: Geometric analysis of bifurcation and symmetry breaking in a Gross-Pitaevskii equation. J. Stat. Phys. 116, 881–905 (2004)MathSciNetzbMATHCrossRefADSGoogle Scholar
  28. 28.
    Kirr E., Kevrekidis P.G., Pelinovsky D.E.: Symmetry-breaking bifurcation in the nonlinear Schrödinger equation with symmetric potentials. Commun. Math. Phys. 308(3), 795–844 (2011)MathSciNetzbMATHCrossRefADSGoogle Scholar
  29. 29.
    Le Coz S., Fukuizumi R., Fibich G., Ksherim B., Sivan Y.: Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential. Phys. D 237(8), 1103–1128 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Marangell R., Jones C.K.R.T., Susanto H.: Localized standing waves in inhomogeneous Schrodinger equations. Nonlinearity 23(9), 2059–2080 (2010)MathSciNetzbMATHCrossRefADSGoogle Scholar
  31. 31.
    Ohta M.: Instability of bound states for abstract nonlinear Schrödinger equations. J. Func. Anal. 261, 90–110 (2011)zbMATHCrossRefGoogle Scholar
  32. 32.
    Pelinovsky, D.E., Phan, T.: Normal form for the symmetry-breaking bifurcation in the nonlinear Schrödinger equation. http://arxiv.org/abs/1101.5402 [nlin.PS], 2011
  33. 33.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics I: Functional Analysis. San Diego, CA: Academic Press Inc., 1980Google Scholar
  34. 34.
    Weinstein M.: Modulational stability of ground states of nonlinear Schrödinger equations. SIAM J. Math. Anal. 16, 472–491 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Weinstein M.: Lyapunov stability of ground states of nonlinear dispersive evolution equations. Comm. Pure Appl. Math. 39, 51–68 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Witthaut, D., Mossmann, S., Korsch, H.J.: Bound and resonance states of the nonlinear Schrödinger equation in simple model systems. J. Phys. A 38, 1777–1702 (2005)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Dipartimento di Scienze MatematichePolitecnico di TorinoTorinoItaly
  2. 2.Dipartimento di Matematica e ApplicazioniUniversità di Milano BicoccaMilanoItaly.

Personalised recommendations