×

On asymptotic stability of moving ground states of the nonlinear Schrödinger equation. (English) Zbl 1293.35289

Summary: We extend to the case of moving solitons, the result on asymptotic stability of ground states of the NLS obtained by the author in an earlier paper. For technical reasons we consider only smooth solutions. The proof is similar to the earlier paper. However now the flows required for the Darboux Theorem and the Birkhoff normal forms, instead of falling within the framework of standard theory of ODE’s, are related to quasilinear hyperbolic symmetric systems. It is also not obvious that the Darboux Theorem can be applied, since we need to compare two symplectic forms in a neighborhood of the ground states not in \( H^{1}(\mathbb{R}^3)\), but rather in the space \( \Sigma \) where also the variance is bounded. But the NLS does not preserve small neighborhoods of the ground states in \( \Sigma \).

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35B40 Asymptotic behavior of solutions to PDEs
35Q51 Soliton equations
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Dario Bambusi and Scipio Cuccagna, On dispersion of small energy solutions to the nonlinear Klein Gordon equation with a potential, Amer. J. Math. 133 (2011), no. 5, 1421 – 1468. · Zbl 1237.35115
[2] Marius Beceanu, A critical center-stable manifold for Schrödinger’s equation in three dimensions, Comm. Pure Appl. Math. 65 (2012), no. 4, 431 – 507. · Zbl 1234.35240
[3] Nabile Boussaid and Scipio Cuccagna, On stability of standing waves of nonlinear Dirac equations, Comm. Partial Differential Equations 37 (2012), no. 6, 1001 – 1056. · Zbl 1251.35098
[4] V. S. Buslaev and G. S. Perel\(^{\prime}\)man, Scattering for the nonlinear Schrödinger equation: states that are close to a soliton, Algebra i Analiz 4 (1992), no. 6, 63 – 102 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 4 (1993), no. 6, 1111 – 1142. · Zbl 0853.35112
[5] V. S. Buslaev and G. S. Perel\(^{\prime}\)man, On the stability of solitary waves for nonlinear Schrödinger equations, Nonlinear evolution equations, Amer. Math. Soc. Transl. Ser. 2, vol. 164, Amer. Math. Soc., Providence, RI, 1995, pp. 75 – 98. · Zbl 0841.35108
[6] Vladimir S. Buslaev and Catherine Sulem, On asymptotic stability of solitary waves for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 20 (2003), no. 3, 419 – 475 (English, with English and French summaries). · Zbl 1028.35139
[7] T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys. 85 (1982), no. 4, 549 – 561. · Zbl 0513.35007
[8] Scipio Cuccagna, The Hamiltonian structure of the nonlinear Schrödinger equation and the asymptotic stability of its ground states, Comm. Math. Phys. 305 (2011), no. 2, 279 – 331. · Zbl 1222.35183
[9] Scipio Cuccagna, On asymptotic stability of ground states of NLS, Rev. Math. Phys. 15 (2003), no. 8, 877 – 903. · Zbl 1084.35089
[10] Scipio Cuccagna, On instability of excited states of the nonlinear Schrödinger equation, Phys. D 238 (2009), no. 1, 38 – 54. · Zbl 1161.35500
[11] Scipio Cuccagna, On scattering of small energy solutions of non-autonomous Hamiltonian nonlinear Schrödinger equations, J. Differential Equations 250 (2011), no. 5, 2347 – 2371. · Zbl 1216.35134
[12] Scipio Cuccagna, On asymptotic stability in energy space of ground states of NLS in 1D, J. Differential Equations 245 (2008), no. 3, 653 – 691. · Zbl 1185.35251
[13] Scipio Cuccagna, Stabilization of solutions to nonlinear Schrödinger equations, Comm. Pure Appl. Math. 54 (2001), no. 9, 1110 – 1145. · Zbl 1031.35129
[14] Scipio Cuccagna and Tetsu Mizumachi, On asymptotic stability in energy space of ground states for nonlinear Schrödinger equations, Comm. Math. Phys. 284 (2008), no. 1, 51 – 77. · Zbl 1155.35092
[15] Scipio Cuccagna, Dmitry Pelinovsky, and Vitali Vougalter, Spectra of positive and negative energies in the linearized NLS problem, Comm. Pure Appl. Math. 58 (2005), no. 1, 1 – 29. · Zbl 1064.35181
[16] Zhou Gang and I. M. Sigal, Relaxation of solitons in nonlinear Schrödinger equations with potential, Adv. Math. 216 (2007), no. 2, 443 – 490. · Zbl 1126.35065
[17] Zhou Gang and Michael I. Weinstein, Dynamics of nonlinear Schrödinger/Gross-Pitaevskii equations: mass transfer in systems with solitons and degenerate neutral modes, Anal. PDE 1 (2008), no. 3, 267 – 322. · Zbl 1175.35136
[18] Manoussos Grillakis, Jalal Shatah, and Walter Strauss, Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal. 74 (1987), no. 1, 160 – 197. · Zbl 0656.35122
[19] Manoussos Grillakis, Jalal Shatah, and Walter Strauss, Stability theory of solitary waves in the presence of symmetry. II, J. Funct. Anal. 94 (1990), no. 2, 308 – 348. · Zbl 0711.58013
[20] E. Kirr and M. I. Weinstein, Diffusion of power in randomly perturbed Hamiltonian partial differential equations, Comm. Math. Phys. 255 (2005), no. 2, 293 – 328. · Zbl 1092.35091
[21] H. P. McKean and J. Shatah, The nonlinear Schrödinger equation and the nonlinear heat equation reduction to linear form, Comm. Pure Appl. Math. 44 (1991), no. 8-9, 1067 – 1080. · Zbl 0773.35075
[22] Edward Nelson, Topics in dynamics. I: Flows, Mathematical Notes, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1969. · Zbl 0197.10702
[23] Peter J. Olver, Applications of Lie groups to differential equations, 2nd ed., Graduate Texts in Mathematics, vol. 107, Springer-Verlag, New York, 1993. · Zbl 0785.58003
[24] Galina Perelman, Asymptotic stability of multi-soliton solutions for nonlinear Schrödinger equations, Comm. Partial Differential Equations 29 (2004), no. 7-8, 1051 – 1095. · Zbl 1067.35113
[25] I. Rodnianski, W. Schlag, A. Soffer, Asymptotic stability of N-soliton states of NLS, (2003), arXiv:math/0309114v1.
[26] Jalal Shatah, Stable standing waves of nonlinear Klein-Gordon equations, Comm. Math. Phys. 91 (1983), no. 3, 313 – 327. · Zbl 0539.35067
[27] A. Soffer and M. I. Weinstein, Multichannel nonlinear scattering for nonintegrable equations, Comm. Math. Phys. 133 (1990), no. 1, 119 – 146. · Zbl 0721.35082
[28] A. Soffer and M. I. Weinstein, Multichannel nonlinear scattering for nonintegrable equations. II. The case of anisotropic potentials and data, J. Differential Equations 98 (1992), no. 2, 376 – 390. · Zbl 0795.35073
[29] A. Soffer and M. I. Weinstein, Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations, Invent. Math. 136 (1999), no. 1, 9 – 74. · Zbl 0910.35107
[30] I. M. Sigal, Nonlinear wave and Schrödinger equations. I. Instability of periodic and quasiperiodic solutions, Comm. Math. Phys. 153 (1993), no. 2, 297 – 320. · Zbl 0780.35106
[31] M. Taylor, Partial differential equations, Appl. Math. Sci. vol. 117, Springer-Verlag, New York, 1996. · Zbl 0869.35001
[32] Tai-Peng Tsai, Asymptotic dynamics of nonlinear Schrödinger equations with many bound states, J. Differential Equations 192 (2003), no. 1, 225 – 282. · Zbl 1038.35128
[33] Tai-Peng Tsai and Horng-Tzer Yau, Asymptotic dynamics of nonlinear Schrödinger equations: resonance-dominated and dispersion-dominated solutions, Comm. Pure Appl. Math. 55 (2002), no. 2, 153 – 216. · Zbl 1031.35137
[34] Tai-Peng Tsai and Horng-Tzer Yau, Relaxation of excited states in nonlinear Schrödinger equations, Int. Math. Res. Not. 31 (2002), 1629 – 1673. · Zbl 1011.35120
[35] Tai-Peng Tsai and Horng-Tzer Yau, Classification of asymptotic profiles for nonlinear Schrödinger equations with small initial data, Adv. Theor. Math. Phys. 6 (2002), no. 1, 107 – 139. · Zbl 1033.81034
[36] Michael I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math. 39 (1986), no. 1, 51 – 67. · Zbl 0594.35005
[37] Michael I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal. 16 (1985), no. 3, 472 – 491. · Zbl 0583.35028
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.