Asymptotic dynamics of nonlinear Schrödinger equations with many bound states. (English) Zbl 1038.35128

The author considers a nonlinear Schrödinger equation with a smooth localized real potential \(V\), \(i\partial_t\psi=(-\Delta+V)\psi+\lambda | \psi| ^2\psi\), where \(-\Delta+V\) is assumed to have three or more bound states. It is shown, under certain conditions on the initial data, that the solution will converge locally to a nonlinear ground state. This extends the author’s previous work with H.-T. Yau [Commun. Pure Appl. Math. 55, 153–216 (2002; Zbl 1031.35137), Int. Math. Res. Notes 31, 1629–1673 (2002; Zbl 1011.35120)].


35Q55 NLS equations (nonlinear Schrödinger equations)
35Q40 PDEs in connection with quantum mechanics
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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[1] Buslaev, V.S.; Perel’man, G.S., Scattering for the nonlinear Schrödinger equations, states close to a soliton, Saint |St. Petersburg math. J., 4, 1111-1142, (1993)
[2] V.S. Buslaev, G.S. Perel’man, On the stability of solitary waves for nonlinear Schrödinger equations, Nonlinear Evolution Equations, Amer. Math. Soc. Translation Series 2, Vol. 164, Amer. Math. Soc., Providence, RI, 1995, pp. 75-98.
[3] V.S. Buslaev, C. Sulem, On the asymptotic stability of solitary waves of nonlinear Schrödinger equations, preprint. · Zbl 1028.35139
[4] Cuccagna, S., Stabilization of solutions to nonlinear Schrödinger equations, Comm. pure appl. math., 54, 9, 1110-1145, (2001) · Zbl 1031.35129
[5] S. Cuccagna, On asymptotic stability of ground states of NLS, preprint. · Zbl 1084.35089
[6] Grillakis, M., Linearized instability for nonlinear Schrödinger and klein – gordon equations, Comm. pure appl. math., 41, 6, 747-774, (1988) · Zbl 0632.70015
[7] Grillakis, M., Analysis of the linearization around a critical point of an infinite dimensional Hamiltonian system, Comm. pure appl. math., 43, 299-333, (1990) · Zbl 0731.35010
[8] Jensen, A.; Kato, T., Spectral properties of Schrödinger operators and time-decay of the wave functions, Duke math. J., 46, 3, 583-611, (1979) · Zbl 0448.35080
[9] Journe, J.-L.; Soffer, A.; Sogge, C.D., Decay estimates for Schrödinger operators, Comm. pure appl. math., 44, 5, 573-604, (1991) · Zbl 0743.35008
[10] Pillet, C.-A.; Wayne, C.E., Invariant manifolds for a class of dispersive, Hamiltonian, partial differential equations, J. differential equations, 141, 310-326, (1997) · Zbl 0890.35016
[11] Rauch, J., Local decay of scattering solutions to Schrödinger’s equation, Comm. math. phys., 61, 2, 149-168, (1978) · Zbl 0381.35023
[12] Shatah, J.; Strauss, W., Instability of nonlinear bound states, Comm. math. phys., 100, 2, 173-190, (1985) · Zbl 0603.35007
[13] J. Shatah, W. Strauss, Spectral condition for instability, Nonlinear PDE’s, Dynamics and Continuum Physics (South Hadley, MA, 1998), Contemporary Mathematics, Vol. 255, Amer. Math. Soc., Providence, RI, 2000, pp. 189-198. · Zbl 0960.47033
[14] Sigal, I.M., Non-linear wave and Schrödinger equations I, instability of periodic and quasiperiodic solutions, Comm. math. phys., 153, 297-320, (1993) · Zbl 0780.35106
[15] Soffer, A.; Weinstein, M.I.; Soffer, A.; Weinstein, M.I., Multichannel nonlinear scattering theory for nonintegrable equations I, II, Comm. math. phys., J. differential equations, 98, 376-390, (1992) · Zbl 0795.35073
[16] Soffer, A.; Weinstein, M.I., Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations, Invent. math., 136, 9-74, (1999) · Zbl 0910.35107
[17] A. Soffer, M.I. Weinstein, Selection of the ground state for nonlinear Schrödinger equations, announcement. · Zbl 1111.81313
[18] Tsai, T.-P.; Yau, H.-T., Asymptotic dynamics of nonlinear Schrödinger equations, resonance dominated and dispersion dominated solutions, Comm. pure appl. math., 55, 153-216, (2002) · Zbl 1031.35137
[19] T.-P. Tsai, H.-T. Yau, Relaxation of excited states in nonlinear Schrödinger equations. Internat. Math. Res. Notes (2002) (31) 1629-1673. · Zbl 1011.35120
[20] Tsai, T.-P.; Yau, H.-T., Stable directions for excited states of nonlinear Schrödinger equations, Comm. partial differential equations, 27, 2363-2402, (2002) · Zbl 1021.35113
[21] Tsai, T.-P.; Yau, H.-T., Classification of asymptotic profiles for nonlinear Schrödinger equations with small initial data, Adv. theor. math. phys., 6, 107-139, (2002) · Zbl 1033.81034
[22] Weinstein, M.I., Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. pure appl. math., 39, 51-68, (1986) · Zbl 0594.35005
[23] Yajima, K., The \(W\^{}\{k,p\}\) continuity of wave operators for Schrödinger operators, J. math. soc. Japan, 47, 3, 551-581, (1995) · Zbl 0837.35039
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