Parametric resonance of ground states in the nonlinear Schrödinger equation

Journal of Differential Equations

Volumn 220, Issue 1, 2006, Pages 85-120

  Cuccagna, S ^a   Kirr, E ^b   Pelinovsky, D ^c  

^a UNIVERSITY OF MODENA AND REGGIO EMILIA   (Italy)

^b UNIVERSITY OF CHICAGO   (United States)

^c MCMASTER UNIVERSITY   (Canada)

Author keywords

Asymptotic stability; Ground states; Nonlinear Schr dinger equation; Parametric resonance

Indexed keywords

EID: 28344456247     PISSN: 00220396     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.jde.2005.07.009     Document Type: Article

References (30)

1. V.S. Buslaev, G.S. Perel'man, On the stability of solitary waves for nonlinear Schrödinger equations. Nonlinear evolution equations, American Mathematical Society Translations Series 2, vol. 164, American Mathematical Society, Providence, RI, 1995, pp. 75-98.
2. T. Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, vol. 10, New York, Courant Institute, 2003.
3. O. Costin R.D. Costin J.L. Lebowitz A. Rokhlenko Evolution of a model quantum system under time periodic forcing: Conditions for complete ionization Comm. Math. Phys. 221 2001 1-26
4. M. Goldberg W. Schlag Dispersive estimates for Schrödinger operators in dimensions one and three Comm. Math. Phys. 251 2004 157-178
5. S. Inouye M.R. Andrews J. Stenger H.J. Miesner D.M. Stemper-Kurn W. Ketterle Observation of Feshbach resonances in a Bose - Einstein condensate Nature 392 1998 151-154
6. A. Jensen R. Kato Spectral properties of Schrödinger operators and time-decay of the wave functions Duke Math. J. 46 1979 583-611
7. P.G. Kevrekidis G. Theocharis D.J. Franzeskakis B.A. Malomed Feshbach resonance management for Bose - Einstein condensates Phys. Rev. Lett. 90 2003 230401
8. E. Kirr M.I. Weinstein Parametrically excited Hamiltonian partial differential equations SIAM J. Math. Anal. 33 2001 16-52
9. E. Kirr M.I. Weinstein Metastable states in parametrically excited multimode Hamiltonian systems Comm. Math. Phys. 236 2003 335-372
10. E. Kirr M.I. Weinstein Diffusion of power in randomly perturbed Hamiltonian partial differential equations Comm. Math. Phys. 255 2005 293-328
11. L. Pitaevskii, S. Stringari, Bose-Einstein condensation, International Series of Monographs on Physics, vol. 116, The Clarendon Press, Oxford University Press, Oxford, 2003.
12. A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer, New York, 1983.
13. D.E. Pelinovsky P.G. Kevrekidis D.J. Franzeskakis Averaging for solitons with nonlinearity management Phys. Rev. Lett. 91 2003 240201
14. D.E. Pelinovsky P.G. Kevrekidis D.J. Franzeskakis V. Zharnitsky Hamiltonian averaging of solitons with nonlinearity management Phys. Rev. E 70 2004 047604
15. n, Applied Mathematical Sciences, vol. 39, Springer, New York, 1984.
16. C.A. Pillet C.E. Wayne Invariant manifolds for a class of dispersive, Hamiltonian, partial differential equations J. Differential Equations 141 1997 310-326
17. H.A. Rose M.I. Weinstein On the bound states of the nonlinear Schrödinger equation with a linear potential Phys. D 30 1-2 1988 207-218
18. I.M. Sigal Nonlinear wave and Schrödinger equations. I: Instability of periodic and quasiperiodic solutions Comm. Math. Phys. 153 1993 297-320
19. B. Simon Pointwise bounds on eigenfunctions and wave packets in N-body quantum systems. I Proc. Amer. Math. Soc. 42 1974 395-401
20. A. Soffer M.I. Weinstein Multichannel nonlinear scattering for nonintegrable equations Comm. Math. Phys. 133 1990 119-146
21. A. Soffer M.I. Weinstein Multichannel nonlinear scattering for nonintegrable equations. II: The case of anisotropic potentials and data J. Differential Equations 98 1992 376-390
22. A. Soffer M.I. Weinstein Nonautonomous Hamiltonians J. Statist. Phys. 93 1998 359-391
23. A. Soffer M.I. Weinstein Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations Invent. Math. 136 1999 9-74
24. A. Soffer M.I. Weinstein Selection of the ground state in nonlinear Schrödinger equations Rev. Math. Phys. 16 2004 977-1071
25. C. Sulem P.L. Sulem The Nonlinear Schrödinger Equation 1999 Springer New York
26. T.P. Tsai H.T. Yau Relaxation of excited states in nonlinear Schrödinger equations Internat. Math. Res. Notices 31 2002 1629-1673
27. K. Yajima Scattering theory for Schrödinger equations with potentials periodic in time J. Math. Soc. Japan 29 1977 729-743
28. p-continuity of wave operators for Schrödinger operators and its applications, in Proceedings of the Korea-Japan Partial Differential Equations Conference (Taejon, 1996), Lecture Notes Series, vol. 39, Seoul, 1997.
29. E. Zeidler Nonlinear Functional Analysis and its Applications. I: Fixed-Point Theorems 1986 Springer New York
30. V. Zharnitsky, D.E. Pelinovsky, Averaging of nonlinearity-managed pulses, Chaos 16 (2006), to be published.