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Scattering in the nonlinear Lamb system. (English) Zbl 1228.81264

Summary: We obtain long time asymptotics for the solutions to a string coupled to a nonlinear oscillator: each finite energy solution decays to a sum of a stationary state and a dispersive wave. The asymptotics hold in global energy norm. The dispersive waves are expressed via initial data and solution to an ordinary differential equation. The asymptotics give a mathematical model for the Bohr’s transitions between quantum stationary states.

MSC:

81U30 Dispersion theory, dispersion relations arising in quantum theory
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
35Q55 NLS equations (nonlinear Schrödinger equations)
34L25 Scattering theory, inverse scattering involving ordinary differential operators
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[1] Bensoussan, A.; Iliine, C.; Komech, A., Arch. ration. mech. anal., 165, 317, (2002)
[2] Buslaev, V.; Komech, A.; Kopylova, E.; Stuart, D., Commun. partial diff. equ., 33, 4, 669, (2008)
[3] Buslaev, V.S.; Perelman, G.S., Saint |St. Petersburg math. J., 4, 1111, (1993)
[4] Buslaev, V.S.; Sulem, C., Ann. inst. Henri Poincaré, anal. non linéaire, 20, 3, 419, (2003)
[5] Cuccagna, S., A survey on asymptotic stability of ground states of nonlinear Schrödinger equations, (), 21-57 · Zbl 1130.35360
[6] Freidlin, M.; Komech, A.I., J. math. phys., 47, 4, 043301, (2006)
[7] Imaikin, V.; Komech, A.; Markowich, P., J. math. phys., 44, 3, 1202, (2003)
[8] Imaikin, V.; Komech, A.; Spohn, H., Russ. J. math. phys., 9, 4, 428, (2002)
[9] Imaikin, V.; Komech, A.; Spohn, H., J. discrete contin. dyn. systems, 10, 1-2, 387, (2003)
[10] Imaikin, V.; Komech, A.; Spohn, H., Monatsh. math., 142, 1-2, 143, (2004)
[11] Imaikin, V.; Komech, A.; Vainberg, B., On scattering of solitons for wave equation coupled to a particle, (), 249-273
[12] Imaikin, V.; Komech, A.; Vainberg, B., Commun. math. phys., 268, 2, 321, (2006)
[13] Keller, J.B.; Bonilla, L.L., Stat. phys., 42, 5-6, 1115, (1986)
[14] Komech, A.I., Moscow univ. math. bull., 46, 34, (1991)
[15] Komech, A.I., J. math. anal. appl., 196, 384, (1995)
[16] Komech, A., Dokl. math., 53, 2, 208, (1996)
[17] Komech, A., Russ. math. surv., 55, 1, 43, (2000)
[18] Komech, A., On Global Attractors of Hamilton Nonlinear Wave Equations, Lecture Notes of the Max Planck Institute for Mathematics in the Sciences, LN 24/2005, Leipzig, 2005
[19] Komech, A.; Kopylova, E., Russian J. math. phys., 13, 2, 158, (2006)
[20] Komech, A.; Kopylova, E.; Stuart, D., On asymptotic stability of solitary waves for schödinger equation coupled to nonlinear oscillator II, SIAM J. Math. Anal. (2008), in press · Zbl 1185.35247
[21] Lamb, H., Proc. London math. soc., 32, 208, (1900)
[22] Merzon, A.; Taneco, M., Phys. lett. A, 372, 4761, (2008)
[23] Morawetz, C.S.; Strauss, W.A., Commun. pure appl. math., 25, 1, (1972)
[24] Reed, M.; Simon, B., Methods of modern mathematical physics, III, (1979), Academic Press · Zbl 0405.47007
[25] Soffer, A.; Weinstein, M.I., J. diff. equ., 98, 376, (1992)
[26] Soffer, A.; Weinstein, M.I., Invent. math., 136, 1, 9, (1999)
[27] Strauss, W.A.; Strauss, W.A., J. funct. anal., J. funct. anal., 43, 281, (1981)
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