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Asymptotic stability of nonlinear Schrödinger equations with potential. (English) Zbl 1086.82013

This paper proves the asymptotic stability of trapped solitons in the generalized nonlinear 1-dimensional Schrödinger equation with a potential, namely, it shows that if an initial condition of the equation is sufficiently close to a trapped soliton, the solution converges to another trapped soliton of the same two-parameter family. This result is obtained under the simplifying restriction of an even potential and even initial conditions, a restriction which according to the authors will be removed in a forthcoming paper. The authors work within the theory of the 1-dimensional Schrödinger operators developed by V. S. Buslaev and G. S. Perelman [J. Math. Sci. 77, 3161–3169 (1995; Zbl 0836.35146)] and V. S. Buslaev and C. Sulem [Ann. Inst. Henri Poincaré Anal. Non Linéaire 20, 419–475 (2003; Zbl 1028.35139)]. This result is an advance with respect to the previous results of A. Softer and M. I. Weinstein [J. Differ. Equations 98, 376–390 (1992; Zbl 0795.35073)] and T-P. Tsai and H.-T. Yau [Commun. Partial Differ. Equations 27, 2363–2402 (2002; Zbl 1021.35113)] which were restricted to the near-linear regime instead of the full nonlinear regime dealt with in this paper.

MSC:

82C20 Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics
35Q55 NLS equations (nonlinear Schrödinger equations)
37K45 Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems
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[1] Ambrosetti A., Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 7 pp 155–
[2] DOI: 10.4310/MRL.2000.v7.n3.a7 · Zbl 0955.35067
[3] Berestycki H., Arch. Ration. Mech. Anal. 82 pp 313–
[4] Buslaev V. S., St. Petersburg Math. J. 4 pp 1111–
[5] DOI: 10.1007/BF02364705 · Zbl 0836.35146
[6] DOI: 10.1016/S0294-1449(02)00018-5 · Zbl 1028.35139
[7] Cazenave T., Textos de Métodos Matemáticos 22, in: An Introduction to Nonlinear Schrödinger Equations (1989)
[8] DOI: 10.1002/cpa.10104 · Zbl 1072.35165
[9] DOI: 10.1002/cpa.20050 · Zbl 1064.35181
[10] DOI: 10.1002/cpa.1018 · Zbl 1031.35129
[11] DOI: 10.1002/1097-0312(200102)54:2<135::AID-CPA1>3.0.CO;2-4 · Zbl 1032.35122
[12] DOI: 10.1142/S0129055X03001849 · Zbl 1084.35089
[13] Cuccagna S., J. Math. Phys. 46
[14] DOI: 10.2307/2946540 · Zbl 0771.35042
[15] Fröhlich J., Comm. Math. Phys. 225 pp 223–
[16] DOI: 10.1016/0022-1236(86)90096-0 · Zbl 0613.35076
[17] Goldstein J. A., Oxford Mathematical Monographs, in: Semigroups of Linear Operators and Applications (1985) · Zbl 0592.47034
[18] Grimshaw R. H. J., Stud. Appl. Math. 97 pp 369–
[19] DOI: 10.1016/0022-1236(87)90044-9 · Zbl 0656.35122
[20] DOI: 10.1016/0022-1236(90)90016-E · Zbl 0711.58013
[21] DOI: 10.1007/978-3-642-55729-3 · Zbl 1033.81004
[22] Hunziker W., J. Math. Phys. 41
[23] Kato T., Perturbation Theory for Linear Operators (1984) · Zbl 0531.47014
[24] DOI: 10.1081/PDE-120002870 · Zbl 0998.35052
[25] DOI: 10.1016/0022-247X(76)90201-8 · Zbl 0333.34020
[26] DOI: 10.1080/03605308808820585 · Zbl 0702.35228
[27] DOI: 10.1007/BF01218621 · Zbl 0693.35132
[28] DOI: 10.1016/0022-0396(89)90123-X · Zbl 0703.35158
[29] Perelman G., Séminaire sur les Équations aux Dérivées Partielles (1995–1996), Exp. No. XIII, Sémin. Équ. Dériv. Partielles, in: Stability of Solitary Waves for Nonlinear Schrödinger Equation (1996)
[30] DOI: 10.1007/BF01609491 · Zbl 0381.35023
[31] Reed M., Methods of Modern Mathematical Physics, I, Functional Analysis (1978)
[32] Reed M., Methods of Modern Mathematical Physics, II, Fourier Analysis (1978)
[33] Reed M., Methods of Modern Mathematical Physics, IV, Analysis of Operators (1978) · Zbl 0401.47001
[34] DOI: 10.1002/cpa.20066 · Zbl 1130.81053
[35] DOI: 10.1007/BF01212446 · Zbl 0603.35007
[36] Sulem C., Applied Mathematical Sciences 139, in: The Nonlinear Schrödinger Equation. Self-Focusing and Wave Collapse (1999) · Zbl 0928.35157
[37] DOI: 10.1007/BFb0035679
[38] DOI: 10.1007/BF02096557 · Zbl 0721.35082
[39] DOI: 10.1016/0022-0396(92)90098-8 · Zbl 0795.35073
[40] DOI: 10.1002/cpa.3012 · Zbl 1031.35137
[41] Tsai T.-P., Int. Math. Res. Not. 31 pp 1629–
[42] DOI: 10.1081/PDE-120016161 · Zbl 1021.35113
[43] DOI: 10.1137/0516034 · Zbl 0583.35028
[44] DOI: 10.1002/cpa.3160390103 · Zbl 0594.35005
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