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On scattering of small energy solutions of non-autonomous Hamiltonian nonlinear Schrödinger equations. (English) Zbl 1216.35134

The author revisits a result by Cuccagna, Kirr and Pelinovsky about the cubic nonlinear Schrödinger equation (NLS) with an attractive localized potential and time-dependent factor in the nonlinearity. He shows that, under generic hypotheses on the linearization at 0 of the equation, small energy solutions are asymptotically free. This is yet a new application of the Hamiltonian structure, continuing a program initiated in a paper by Bambusi and Cuccagna.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35P25 Scattering theory for PDEs
37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)
35B44 Blow-up in context of PDEs
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References:

[1] D. Bambusi, S. Cuccagna, On dispersion of small energy solutions of the nonlinear Klein Gordon equation with a potential, Amer. J. Math., in press, arXiv:0908.4548. · Zbl 1237.35115
[2] Buslaev, V.; Perelman, G., On the stability of solitary waves for nonlinear Schrödinger equations, (), 75-98 · Zbl 0841.35108
[3] Cuccagna, S., The Hamiltonian structure of the nonlinear Schrödinger equation and the asymptotic stability of its ground states · Zbl 1222.35183
[4] Cuccagna, S., Orbitally but not asymptotically stable ground states for the discrete NLS, Discrete contin. dyn. syst. ser. A, 238, 105-134, (2010) · Zbl 1180.35479
[5] Cuccagna, S., On instability of excited states of the nonlinear Schrödinger equation, Phys. D, 238, 38-54, (2009) · Zbl 1161.35500
[6] Cuccagna, S.; Pelinovsky, D.; Kirr, E.W., Parametric resonance of ground states in the nonlinear Schrödinger equation, J. differential equations, 220, 85-120, (2006) · Zbl 1081.35101
[7] Cuccagna, S.; Mizumachi, T., On asymptotic stability in energy space of ground states for nonlinear Schrödinger equations, Comm. math. phys., 284, 51-87, (2008) · Zbl 1155.35092
[8] Cuccagna, S.; Tarulli, M., On asymptotic stability of standing waves of discrete Schrödinger equation in Z, SIAM J. math. anal., 41, 861-885, (2009) · Zbl 1189.35303
[9] Gang, Zhou, Perturbation expansion and n-th order Fermi Golden rule of the nonlinear Schrödinger equations, J. math. phys., 48, 053509, (2007), 23 pp · Zbl 1144.81430
[10] Gang, Zhou; Sigal, I.M., Relaxation of solitons in nonlinear Schrödinger equations with potential, Adv. math., 216, 443-490, (2007) · Zbl 1126.35065
[11] Gang, Zhou; Weinstein, M.I., Dynamics of nonlinear Schrödinger/Gross-pitaeskii equations; mass transfer in systems with solitons and degenerate neutral modes, Anal. PDE, 1, 267-322, (2008) · Zbl 1175.35136
[12] Kirr, E.; Weinstein, M.I., Diffusion of power in randomly perturbed Hamiltonian partial differential equations, Comm. math. phys., 255, 2, 293-328, (2005) · Zbl 1092.35091
[13] Mizumachi, T., Asymptotic stability of small solitons to 1D NLS with potential, J. math. Kyoto univ., 48, 471-497, (2008) · Zbl 1175.35138
[14] Mizumachi, T., Asymptotic stability of small solitons for 2D nonlinear Schrödinger equations with potential, J. math. Kyoto univ., 43, 599-620, (2007) · Zbl 1146.35085
[15] Reed, M.; Simon, B., Methods of modern mathematical physics, (1978), Academic Press San Diego
[16] Sigal, I.M., Nonlinear wave and Schrödinger equations. I. instability of periodic and quasi-periodic solutions, Comm. math. phys., 153, 297-320, (1993) · Zbl 0780.35106
[17] Soffer, A.; Weinstein, M.I., Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations, Invent. math., 136, 9-74, (1999) · Zbl 0910.35107
[18] Tsai, T.P., Asymptotic dynamics of nonlinear Schrödinger equations with many bound states, J. differential equations, 192, 225-282, (2003) · Zbl 1038.35128
[19] Yajima, K., Resonances for the AC-Stark effect, Comm. math. phys., 87, 331-352, (1982) · Zbl 0538.47010
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