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Soliton dynamics for the nonlinear Schrödinger equation with magnetic field

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Abstract

The semiclassical regime of a nonlinear focusing Schrödinger equation in presence of non-constant electric and magnetic potentials V, A is studied by taking as initial datum the ground state solution of an associated autonomous stationary equation. The concentration curve of the solutions is a parameterization of the solutions of the second order ordinary equation \({\ddot x=-\nabla V(x)-\dot x\times B(x)}\), where \({B=\nabla\times A}\) is the magnetic field of a given magnetic potential A.

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References

  1. Abou Salem W.K.: Solitary wave dynamics in time-dependent potentials. J. Math. Phys. 49, 032101 (2008)

    Article  MathSciNet  Google Scholar 

  2. Ambrosetti A., Malchiodi A.: Perturbation Methods and Semilinear Elliptic Problems on \({\mathbb{R}^n}\), Progress in Mathematics, vol. 240, pp. xii+183. Birkhäuser Verlag, Basel (2006)

    Google Scholar 

  3. Arioli G., Szulkin A.: A semilinear Schrödinger equations in the presence of a magnetic field. Arch. Ration. Mech. Anal. 170, 277–295 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Avron J.E., Herbst I., Simon B.: Schrödinger operators with magnetic fields. I. General interactions. Duke Math. J. 45, 847–883 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  5. Avron J.E., Herbst I.W., Simon B.: Separation of center of mass in homogeneous magnetic fields. Ann. Phys. 114, 431–451 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  6. Avron J.E., Herbst I.W., Simon B.: Schrödinger operators with magnetic fields. III. Atoms in homogeneous magnetic field. Commun. Math. Phys. 79, 529–572 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bartsch T., Dancer E.N., Peng S.: On multi-bump semi-classical bound states of nonlinear Schrödinger equations with electromagnetic fields. Adv. Differ. Equ. 11, 781–812 (2006)

    MathSciNet  MATH  Google Scholar 

  8. Beresticki H., Lions P.L.: Nonlinear scalar fields equation I. Existence of a ground state. Arch. Ration. Mech. Anal. 82, 313–346 (1983)

    Google Scholar 

  9. Bronski J., Jerrard R.: Soliton dynamics in a potential. Math. Res. Lett. 7, 329–342 (2000)

    MathSciNet  MATH  Google Scholar 

  10. Buslaev, V.S., Perelman, G.S.: On the Stability of Solitary Waves for Nonlinear Schrödinger Equations. Nonlinear Evolution Equations. American Mathematical Society Translational Series 2, vol. 164, pp. 75–98, American Mathematical Society, Providence, RI (1995)

  11. Buslaev V.S., Perelman G.S.: Scattering for the nonlinear Schrödinger equation: states that are close to a soliton. Algebra i Analiz. 4, 63–102 (1992)

    MathSciNet  MATH  Google Scholar 

  12. Buslaev V.S., Sulem C.: On asymptotic stability of solitary waves for nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 20, 419–475 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. Carles R.: Nonlinear Schrödinger equations with repulsive harmonic potential and applications. SIAM J. Math. Anal. 35, 823–843 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Carles R.: WKB analysis for nonlinear Schrödinger equations with potential. Commun. Math. Phys. 269, 195–221 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Cazenave, T.: An Introduction to Nonlinear Schrödinger equation, Textos de Métodos Matemd́fticos, vol. 26, Federal University of Rio de Janeiro, Rio de Janeiro (1993)

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

    Article  MathSciNet  MATH  Google Scholar 

  17. Cazenave T., Weissler F.B.: The Cauchy problem for the nonlinear Schrödinger equation in H 1. Manuscr. Math. 61, 477–494 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  18. Chabrowski J., Szulkin A.: On the Schrödinger equation involving a critical Sobolev exponent and magnetic field. Topol. Method. Nonlinear Anal. 25, 3–21 (2005)

    MathSciNet  MATH  Google Scholar 

  19. Cingolani S.: Semiclassical stationary states of nonlinear Schrödinger equations with an external magnetic field. J. Differ. Equ. 188, 52–79 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. Cingolani S., Secchi S.: Semiclassical limit for nonlinear Schrödinger equations with electromagnetic fields. J. Math. Anal. Appl. 275, 108–130 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  21. Cingolani S., Jeanjean L., Secchi S.: Multi-peak solutions for magnetic NLS equations without non-degeneracy conditions. ESAIM-COCV. 15, 653–675 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. D’Ancona P., Fanelli L.: Strichartz and smoothing estimates of dispersive equations with magnetic potentials. Comm. Partial Differ. Equ. 33, 1082–1112 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  23. Esteban M.J., Lions P.L.: Stationary Solutions of Nonlinear Schrödinger Equations with an External Magnetic Field. In: Colombin F et al. (eds.) PDE and Calculus of Variations, Essays in Honor of E. De Giorgi. Birkhäuser, Boston (1990)

    Google Scholar 

  24. Floer A., Weinstein A.: Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal. 69, 397–408 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  25. Fröhlich J., Gustafson S., Jonsson B.L.G., Sigal I.M.: Dynamics of solitary waves external potentials. Commun. Math. Phys. 250, 613–642 (2004)

    Article  MATH  Google Scholar 

  26. Fröhlich J., Tsai T.-P., Yau H.-T.: On the point-particle (Newtonian) limit of the non-linear Hartree equation. Commun. Math. Phys. 225, 223–274 (2002)

    Article  MATH  Google Scholar 

  27. Fröhlich, J., Tsai, T.-P., Yau, H.-T.: On a classical limit of quantum theory and the non-linear Hartree equation. GAFA 2000 (Tel Aviv, 1999). Geom. Funct. Anal., Special Volume, Part I, pp. 57–78 (2000)

  28. Fröhlich, J., Tsai, T.-P., Yau, H.-T.: On a classical limit of quantum theory and the non-linear Hartree equation. Conference Mosh Flato 1999, vol. I (Dijon), pp. 189–207; Math. Phys. Stud. vol. 21, Kluwer Academic Publisher, Dordrecht (2000)

  29. Gilbarg D., Trudinger N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edn, vol. 224, pp xiii+513. Springer-Verlag, Berlin (1983)

    Google Scholar 

  30. Grillakis M., Shatah J., Strauss W.: Stability theory of solitary waves in the presence of symmetry. I. J. Funct. Anal. 74, 160–197 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  31. Grillakis M., Shatah J., Strauss W.: Stability theory of solitary waves in the presence of symmetry. II. J. Funct. Anal. 94, 308–348 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  32. Gustafson S., Sigal I.M.: Effective dynamics of magnetic vortices. Adv. Math. 199, 448–498 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  33. Holmer, J., Zworski, M.: Soliton interaction with slowly varying potentials, Int. Math. Res. Not. IMRN, vol. 10, pp 36 (2008)

  34. Holmer J., Zworski M.: Slow soliton interaction with delta impurities. J. Mod. Dyn. 1, 689–718 (2007)

    MathSciNet  MATH  Google Scholar 

  35. Jonsson B.L.G., Fröhlich J., Gustafson S., Sigal I.M.: Long time motion of NLS solitary waves in a confining potential. Ann. Henri Poincaré 7, 621–660 (2006)

    Article  MATH  Google Scholar 

  36. Kaup D.J., Newell A.C.: Solitons as particles and oscillators and in slowly changing media: a singular perturbation theory. Proc. R. Soc. Lond. A. 361, 413–446 (1978)

    Article  Google Scholar 

  37. Keener J.P., McLaughlin D.W.: Solitons under perturbation. Phys. Rev. A 16, 777–790 (1977)

    Article  MathSciNet  Google Scholar 

  38. Keraani S.: Semiclassical limit of a class of Schrödinger equation with potential. Comm. Partial Differ. Equ. 27, 693–704 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  39. Keraani S.: Semiclassical limit for nonlinear Schrödinger equation with potential. II. Asymptot. Anal. 47, 171–186 (2006)

    MathSciNet  MATH  Google Scholar 

  40. Kurata K.: Existence and semi-classical limit of the least energy solution to a nonlinear Schrödinger equation with electromagnetic fields. Nonlinear Anal. 41, 763–778 (2000)

    Article  MathSciNet  Google Scholar 

  41. Lions P.L.: The concentration-compactness principle in the calculus of variations. The locally compact case. Part II. Annales Inst. H. Poincaré Anal. Nonlinear 1, 223–283 (1984)

    MATH  Google Scholar 

  42. Michel L.: Remarks on nonlinear Schrödinger equation with magnetic fields. Commun. Partial Differ. Equ. 33, 1198–1215 (2008)

    Article  MATH  Google Scholar 

  43. Nakamura Y., Shimomura A.: Local well-posedness and smoothing effects of strong solutions for nonlinear Schrödinger equations with potentials and magnetic fields. Hokkaido Math. J. 34, 37–63 (2005)

    MathSciNet  MATH  Google Scholar 

  44. Nakamura Y.: Local solvability and smoothing effects of nonlinear Schrödinger equations with magnetic fields. Funkcial Ekvac. 44, 1–18 (2001)

    MathSciNet  MATH  Google Scholar 

  45. Reed M., Simon B.: Methods of Modern Mathematical Physics. I. Functional Analysis. 2nd edn, pp. xv+400. Academic Press, Inc., New York (1980)

    MATH  Google Scholar 

  46. Rodnianski I., Schlag W., Soffer A.: Dispersive analysis of charge transfer models. Commun. Pure Appl. Math. 58, 149–216 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  47. Secchi S., Squassina M.: On the location of spikes for the Schrödinger equation with electromagnetic field. Commun. Contemp. Math. 7, 251–268 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  48. Selvitella A.: Asymptotic evolution for the semiclassical nonlinear Schrödinger equation in presence of electric and magnetic fields. J. Differ. Equ. 245, 2566–2584 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  49. Simon B.: Functional Integration and Quantum Physics. Pure and Applied Mathematics, vol. 86, pp. ix+296. Academic Press, Inc., New York (1979)

    Google Scholar 

  50. Soffer A., Weinstein M.I.: Multichannel nonlinear scattering for nonintegrable equations. Commun. Math. Phys. 133, 119–146 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  51. Soffer A., Weinstein M.I.: Multichannel nonlinear scattering for nonintegrable equations. II. The case of anisotropic potentials and data. J. Differ. Equ. 98, 376–390 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  52. Soffer A., Weinstein M.I.: Selection of the ground state for nonlinear Schrödinger equations. Rev. Math. Phys. 16, 977–1071 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  53. Sulem C., Sulem P.L.: The Nonlinear Schrödinger Equation. Self-Focusing and Wave Collapse. Applied Mathematical Sciences, vol. 139, pp +350. Springer-Verlag, New York (1999)

    Google Scholar 

  54. Tao T.: Why are solitons stable?. Bull. Am. Math. Soc. 46, 1–33 (2009)

    Article  MATH  Google Scholar 

  55. Tsai T.-P., Yau H.-T.: Asymptotic dynamics of nonlinear Schrödinger equations: resonance-dominated and dispersion-dominated solutions. Commun. Pure Appl. Math. 55, 153–216 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  56. Tsai T.-P., Yau H.-T.: Relaxation of excited states in nonlinear Schrödinger equations. Int. Math. Res. Not. 31, 1629–1673 (2002)

    Article  MathSciNet  Google Scholar 

  57. Tsai T.-P., Yau H.-T.: Stable directions for excited states of nonlinear Schrödinger equations. Commun. Partial Differ. Equ. 27, 2363–2402 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  58. Weinstein M.: Modulation stability of ground state of nonlinear Schrödinger equations. SIAM J. Math. Anal. 16, 472–491 (1985)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  60. Yajima K.: Schrödinger evolution equations with magnetic fields. J. d’Analyse Math. 56, 29–76 (1991)

    Article  MathSciNet  MATH  Google Scholar 

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  1. Department of Computer Science, University of Verona, Cà Vignal 2, Strada Le Grazie 15, 37134, Verona, Italy

    Marco Squassina

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Squassina, M. Soliton dynamics for the nonlinear Schrödinger equation with magnetic field. manuscripta math. 130, 461–494 (2009). https://doi.org/10.1007/s00229-009-0307-y

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