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Soliton dynamics for the nonlinear Schrödinger equation with magnetic field. (English) Zbl 1179.81066

Summary: 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 \(A\).

MSC:

81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35Q40 PDEs in connection with quantum mechanics
35Q51 Soliton equations
35Q55 NLS equations (nonlinear Schrödinger equations)
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
37K45 Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems
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[1] Abou Salem W.K.: Solitary wave dynamics in time-dependent potentials. J. Math. Phys. 49, 032101 (2008) · Zbl 1153.81428
[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) · Zbl 1115.35004
[3] Arioli G., Szulkin A.: A semilinear Schrödinger equations in the presence of a magnetic field. Arch. Ration. Mech. Anal. 170, 277–295 (2003) · Zbl 1051.35082
[4] Avron J.E., Herbst I., Simon B.: Schrödinger operators with magnetic fields. I. General interactions. Duke Math. J. 45, 847–883 (1978) · Zbl 0399.35029
[5] Avron J.E., Herbst I.W., Simon B.: Separation of center of mass in homogeneous magnetic fields. Ann. Phys. 114, 431–451 (1978) · Zbl 0409.35027
[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) · Zbl 0464.35086
[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) · Zbl 1146.35081
[8] Beresticki H., Lions P.L.: Nonlinear scalar fields equation I. Existence of a ground state. Arch. Ration. Mech. Anal. 82, 313–346 (1983)
[9] Bronski J., Jerrard R.: Soliton dynamics in a potential. Math. Res. Lett. 7, 329–342 (2000) · Zbl 0955.35067
[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) · Zbl 0841.35108
[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) · Zbl 0795.35111
[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) · Zbl 1028.35139
[13] Carles R.: Nonlinear Schrödinger equations with repulsive harmonic potential and applications. SIAM J. Math. Anal. 35, 823–843 (2003) · Zbl 1054.35090
[14] Carles R.: WKB analysis for nonlinear Schrödinger equations with potential. Commun. Math. Phys. 269, 195–221 (2007) · Zbl 1123.35062
[15] Cazenave, T.: An Introduction to Nonlinear Schrödinger equation, Textos de Métodos Matemdfticos, 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) · Zbl 0513.35007
[17] Cazenave T., Weissler F.B.: The Cauchy problem for the nonlinear Schrödinger equation in H 1. Manuscr. Math. 61, 477–494 (1988) · Zbl 0696.35153
[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) · Zbl 1176.35022
[19] Cingolani S.: Semiclassical stationary states of nonlinear Schrödinger equations with an external magnetic field. J. Differ. Equ. 188, 52–79 (2003) · Zbl 1062.81056
[20] Cingolani S., Secchi S.: Semiclassical limit for nonlinear Schrödinger equations with electromagnetic fields. J. Math. Anal. Appl. 275, 108–130 (2002) · Zbl 1014.35087
[21] Cingolani S., Jeanjean L., Secchi S.: Multi-peak solutions for magnetic NLS equations without non-degeneracy conditions. ESAIM-COCV. 15, 653–675 (2009) · Zbl 1221.35393
[22] D’Ancona P., Fanelli L.: Strichartz and smoothing estimates of dispersive equations with magnetic potentials. Comm. Partial Differ. Equ. 33, 1082–1112 (2008) · Zbl 1160.35363
[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)
[24] Floer A., Weinstein A.: Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal. 69, 397–408 (1986) · Zbl 0613.35076
[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) · Zbl 1075.35075
[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) · Zbl 1025.81015
[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) · Zbl 1050.81015
[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) · Zbl 1071.81543
[29] Gilbarg D., Trudinger N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edn, vol. 224, pp xiii+513. Springer-Verlag, Berlin (1983) · Zbl 0562.35001
[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) · Zbl 0656.35122
[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) · Zbl 0711.58013
[32] Gustafson S., Sigal I.M.: Effective dynamics of magnetic vortices. Adv. Math. 199, 448–498 (2006) · Zbl 1081.35102
[33] Holmer, J., Zworski, M.: Soliton interaction with slowly varying potentials, Int. Math. Res. Not. IMRN, vol. 10, pp 36 (2008) · Zbl 1147.35084
[34] Holmer J., Zworski M.: Slow soliton interaction with delta impurities. J. Mod. Dyn. 1, 689–718 (2007) · Zbl 1137.35060
[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) · Zbl 1100.81019
[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)
[37] Keener J.P., McLaughlin D.W.: Solitons under perturbation. Phys. Rev. A 16, 777–790 (1977)
[38] Keraani S.: Semiclassical limit of a class of Schrödinger equation with potential. Comm. Partial Differ. Equ. 27, 693–704 (2002) · Zbl 0998.35052
[39] Keraani S.: Semiclassical limit for nonlinear Schrödinger equation with potential. II. Asymptot. Anal. 47, 171–186 (2006) · Zbl 1133.35092
[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) · Zbl 0993.35081
[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) · Zbl 0704.49004
[42] Michel L.: Remarks on nonlinear Schrödinger equation with magnetic fields. Commun. Partial Differ. Equ. 33, 1198–1215 (2008) · Zbl 1159.35068
[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) · Zbl 1067.35111
[44] Nakamura Y.: Local solvability and smoothing effects of nonlinear Schrödinger equations with magnetic fields. Funkcial Ekvac. 44, 1–18 (2001) · Zbl 1143.35369
[45] Reed M., Simon B.: Methods of Modern Mathematical Physics. I. Functional Analysis. 2nd edn, pp. xv+400. Academic Press, Inc., New York (1980) · Zbl 0459.46001
[46] Rodnianski I., Schlag W., Soffer A.: Dispersive analysis of charge transfer models. Commun. Pure Appl. Math. 58, 149–216 (2005) · Zbl 1130.81053
[47] Secchi S., Squassina M.: On the location of spikes for the Schrödinger equation with electromagnetic field. Commun. Contemp. Math. 7, 251–268 (2005) · Zbl 1157.35482
[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) · Zbl 1154.35084
[49] Simon B.: Functional Integration and Quantum Physics. Pure and Applied Mathematics, vol. 86, pp. ix+296. Academic Press, Inc., New York (1979) · Zbl 0434.28013
[50] Soffer A., Weinstein M.I.: Multichannel nonlinear scattering for nonintegrable equations. Commun. Math. Phys. 133, 119–146 (1990) · Zbl 0721.35082
[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) · Zbl 0795.35073
[52] Soffer A., Weinstein M.I.: Selection of the ground state for nonlinear Schrödinger equations. Rev. Math. Phys. 16, 977–1071 (2004) · Zbl 1111.81313
[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) · Zbl 0928.35157
[54] Tao T.: Why are solitons stable?. Bull. Am. Math. Soc. 46, 1–33 (2009) · Zbl 1155.35082
[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) · Zbl 1031.35137
[56] Tsai T.-P., Yau H.-T.: Relaxation of excited states in nonlinear Schrödinger equations. Int. Math. Res. Not. 31, 1629–1673 (2002) · Zbl 1011.35120
[57] Tsai T.-P., Yau H.-T.: Stable directions for excited states of nonlinear Schrödinger equations. Commun. Partial Differ. Equ. 27, 2363–2402 (2002) · Zbl 1021.35113
[58] Weinstein M.: Modulation stability of ground state of nonlinear Schrödinger equations. SIAM J. Math. Anal. 16, 472–491 (1985) · Zbl 0583.35028
[59] Weinstein M.: Lyapunov stability of ground states of nonlinear dispersive evolution equations. Commun. Pure Appl. Math. 39, 51–67 (1986) · Zbl 0594.35005
[60] Yajima K.: Schrödinger evolution equations with magnetic fields. J. d’Analyse Math. 56, 29–76 (1991) · Zbl 0739.35083
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