Asymptotic stability of nonlinear Schrödinger equations with potential

Reviews in Mathematical Physics

Volumn 17, Issue 10, 2005, Pages 1143-1207

  Gang, Zhou ^a   Sigal, I M ^a  

^a UNIVERSITY OF TORONTO   (Canada)

Author keywords

Asymptotic stability; Nonlinear Schr dinger equations; Potential; Soliton

Indexed keywords

EID: 27944461890     PISSN: 0129055X     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0129055X05002522     Document Type: Article

References (51)

1. A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations with bounded potentials, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 7(3) (1996) 155-160.
2. J. C. Bronski and R. L. Jerrard, Soliton dynamics in a potential, Math. Res. Lett. 7(2, 3) (2000) 329-342.
3. H. Berestycki and P.-L. Lions, Nonlinear scalar field equations, I, Existence of a ground state, Arch. Ration. Mech. Anal. 82(4) (1983) 313-345.
4. V. S. Buslaev and G. S. Perelman, Scattering for the nonlinear Schrödinger equation: States close to a soliton, St. Petersburg Math. J. 4(6) (1993) 1111-1142.
5. V. S. Buslaev and G. S. Perelman, Nonlinear scattering: The states which are close to a soliton, J. Math. Sci. 77(3) (1995) 3161-3169.
6. V. S. Buslaev and C. Sulem, On asymptotic stability of solitary waves for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linēaire 20(3) (2003) 419-475.
7. T. Cazenave, An Introduction to Nonlinear Schrödinger Equations, Textos de Métodos Matemáticos, Vol. 22 (I.M.U.F.R.J., Rio de Janeiro, 1989).
8. A. Comech, On orbital stability of quasistationary solitary waves of minimal energy, preprint.
9. A. Comech and D. Pelinovsky, Purely nonlinear instability of standing waves with minimal energy, Comm. Pure Appl. Math. 56(11) (2003) 1565-1607.
10. S. Cuccagna, D. Pelinovsky and V. Vougalter, Spectra of positive and negative energies in the linearized NLS problem, Comm. Pure Appl. Math. 58(1) (2005) 1-29.
11. S. Cuccagna, Stabilization of solutions to nonlinear Schrödinger equations, Comm. Pure Appl. Math. 54(9) (2001) 1110-1145.
12. S. Cuccagna, On asymptotic stability of ground states of NLS, Comm. Pure Appl. Math. 54(2) (2001) 135-152.
13. S. Cuccagna, On asymptotic stability of ground states of NLS, Rev. Math. Phys. 15(8) (2003) 877-903.
14. S. Cuccagna and D. Pelinovsky, Bifurcations from the end points of the essential spectrum in the linearized NLS problem, J. Math. Phys. 46 (2005).
15. P. Deift and X. Zhou, A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKDV equation, Ann. of Math. (2) 137(2) (1993) 295-368.
16. J. Fröhlich, S. Gustafson, B. L. G. Jonsson and I. M. Sigal, Solitary wave dynamics in an external potential, Comm. Math. Phys. (to appear).
17. J. Fröhlich, T. P. Tsai and H. T. Yau, On the point-particle (Newtonian) limit of the nonlinear Hartree equation, Comm. Math. Phys. 225(2) (2002) 223-274.
18. A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal. 69 (1986) 397-408.
19. J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford Mathematical Monographs (Oxford University Press, New York, 1985).
20. M. Goldberg and W. Schlag, Dispersive estimates for Schrödinger operators in dimensions one and three, arXiv:math.AP/0306108 v1 2003.
21. R. H. J. Grimshaw and D. Pelinovsky, Nonlocal models for envelope waves in a stratified fluid, Stud. Appl. Math. 97(4) (1996) 369-391.
22. M. Grillalds, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry, I, J. Funct. Anal. 74(1) (1987) 160-197.
23. M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry, II, J. Funct. Anal. 94(2) (1990) 308-348.
24. S. J. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics (Springer, New York, 2003).
25. W. Hunziker and I. M. Sigal, The quantum N-body problem, J. Math. Phys. 41(6) (2000).
26. B. L. G. Jonsson, M. Merkli and I. M. Sigal, Lectures in Applied Analysis, preliminary text, Toronto (2005), www.math.toronto.edu/sigal.
27. T. Kato, Perturbation Theory for Linear Operators (Springer-Verlag, Berlin/New York, 1984).
28. S. Keraani, Semiclassical limit of a class of Schrödinger equations with potential, Commun. Partial Differential Equations 27(3, 4) (2002) 693-704.
29. D. J. Kaup, Closure of the squared Zakharov-Shabat eigenstates, J. Math. Anal. Appl. 54(3) (1976) 849-864.
30. a), Commun. Partial Differential Equations 13(12) (1988) 1499-1519.
31. Y.-G. Oh, Stability of semiclassical bound states of nonlinear Schrödinger equations with potentials, Commun. Math. Phys. 121 (1989) 11-33.
32. Y.-G. Oh, Cauchy problem and Ehrenfest's law of nonlinear Schrödinger equations with potential, J. Differential Equations 81 (1989) 255-274.
33. G. Perelman, Stability of Solitary Waves for Nonlinear Schrödinger Equation, Séminaire sur les Équations aux Dérivées Partielles (1995-1996), Exp. No. XIII, Sémin. Équ. Dériv. Partielles (École Polytech., Palaiseau, 1996).
34. J. Rauch, Local decay of scattering solutions to Schrödinger eqaution, Comm. Math. Phys. 61 (1978) 149-168.
35. M. Reed and B. Simon, Methods of Modern Mathematical Physics, I, Functional Analysis (Academic Press, 1978).
36. M. Reed and B. Simon, Methods of Modern Mathematical Physics. II, Fourier Analysis (Academic Press, 1978).
37. M. Reed and B. Simon, Methods of Modern Mathematical Physics, IV, Analysis of Operators (Academic Press, 1978).
38. I. Rodnianski, W. Schlag and A. Soffer, Dispersive analysis of charge transfer models, Comm. Pure Appl. Math. 58(2) (2005) 149-216.
39. I. Rodnianski, W. Schlag and A. Soffer, Asymptotic stability of N-soliton state of NLS, arXivimath.AP.
40. W. Schlag, Stable manifold for orbitally unstable NLS, arXivimath.AP.
41. J. Shatah and W. Strauss, Instability of nonlinear bound states, Comm. Math. Phys. 100(2) (1985) 173-190.
42. C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation. Self-Focusing and Wave Collapse, Applied Mathematical Sciences, Vol. 139 (Springer-Verlag, New York, 1999).
43. A. Soffer and M. I. Weinstein, Multichannel nonlinear scattering for nonintegrable equations, in Integrable Systems and Applications (Île d'Oléron, 1988) Lecture Notes in Physics, Vol. 342 (Springer, Berlin, 1989), pp. 312-327.
44. A. Soffer and M. I. Weinstein, Multichannel nonlinear scattering for nonintegrable equations, Comm. Math. Phys. 133 (1990) 119-146.
45. A. Soffer and M. I. Weinstein, Multichannel nonlinear scattering for nonintegrable equations. II. The case of anisotropic potentials and data, J. Differential Equations 98(2) (1992) 376-390.
46. A. Soffer and M. I. Weinstien, Selection of the ground state for nonlinear Schrödinger equations, to appear in Rev. Math. Phys.
47. T.-P. Tsai and H.-T. Yau, Asymptotic dynamics of nonlinear Schrödinger equations: Resonance-dominated and dispersion-dominated solutions, Comm. Pure. Appl. Math. 55 (2002) 153-216.
48. T.-P. Tsai and H.-T. Yau, Relaxation of excited states in nonlinear Schrödinger equations, Int. Math. Res. Not. 31 (2002) 1629-1673.
49. T.-P. Tsai and H.-T. Yau, Stable directions for excited states of nonlinear Schrödinger equations, Comm. Partial Differential Equations 27(11, 12) (2002) 2363-2402.
50. M. I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal. 16(3) (1985) 472-491.
51. M. I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math. 39(1) (1986) 51-67.