Perturbation expansion and Nth order Fermi golden rule of the nonlinear Schrödinger equations

Journal of Mathematical Physics

Volumn 48, Issue 5, 2007, Pages

  Zhou, Gang ^a  

^a UNIVERSITY OF TORONTO   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 34249902487     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.2716971     Document Type: Article

References (30)

1. Ambrosetti, A., Badiale, M., and Cingolani, S., " Semiclassical states of nonlinear Schrödinger equations with bounded potentials., " Atti Accad. Naz. Lincei, Cl. Sci. Fis., Mat. Nat., Rend. Lincei, Mat. Appl. 7, 155-160 (1996).
2. Buslaev, V. S., and Perelman, G. S., " Scattering for the nonlinear Schrödinger equation: States close to a soliton., " St. Petersbg. Math. J. 4, 1111-1142 (1993).
3. Buslaev, V. S., and Perelman, G. S., " Nonlinear scattering: the states which are close to a soliton., " J. Math. Sci. (N.Y.) 77, 3161-3169 (1995).
4. Buslaev, V., and Sulem, C., " On asymptotic stability of solitary waves for nonlinear Schrödinger equations., " Ann. Inst. Henri Poincare, Anal. Non Lineaire 20, 419-475 (2003).
5. Berestycki, H., and Lions, P.-L., " Nonlinear scalar field equations, I. Existence of a ground states., " Arch. Ration. Mech. Anal. 82, 313-345 (1983).
6. Cazenave, T., An Introduction to Nonlinear Schrödinger Equations (American Mathematical Society, Providence, 2003).
7. Cuccagna, S., and Pelinovsky, D., " Bifurcations from the endpoints of the essential spectrum in the linearized nonlinear Schrödinger problem., " J. Math. Phys. 46, 053520 (2005).
8. Cuccagna, S., Pelinovsky, D., and Vougalter, V., " Spectra of positive and negative energies in the linearized NLS problem., " Commun. Pure Appl. Math. 58, 1-29 (2005).
9. Cuccagna, S., " Stabilization of solutions to nonlinear Schrödinger equations., " Commun. Pure Appl. Math. 54, 1110-1145 (2001).
10. Dirac, P. A. M., " The quantum theory of the emission and absorption of radiation., " Proc. R. Soc. London, Ser. A 114, 243-265 (1927).
11. Floer, A., and Weinstein, A., " Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential., " J. Funct. Anal. 69, 397-408 (1986).
12. Gang, Z., and Sigal, I. M., " Asymptotic stability of trapped solitons of nonlinear Schrödinger equations with potential., " Rev. Math. Phys. 17, 1143-1207 (2005).
13. Gang, Z., and Sigal, I. M., " On soliton dynamics in nonlinear Schrödinger equations., " Geom. Funct. Anal. 16, No. 7, 1377-1390 (2006).
14. Gang, Z., and Sigal, I. M., " Relaxation to trapped solitons in nonlinear Schrödinger equations with potential., " to appear in Advances in Mathematics.
15. Grillakis, M., Shatah, J., and Strauss, W., " Stability theory of solitary waves in the presence of symmetry. I.., " J. Funct. Anal. 74, 160-197 (1987).
16. Oh, Y.-G., " Existence of semiclassical bound states of nonlinear Schrödinger equations with potential of the class (Va)., " Commun. Partial Differ. Equ. 13, 1499-1519 (1988).
17. Oh, Y.-G., " Stability of semiclassical bound states of nonlinear Schrödinger equations with potentials., " Commun. Math. Phys. 121, 11-33 (1989).
18. Sigal, I. M., " Nonlinear wave and Schrödinger equations. I. Instability of periodic and quasiperiodic solutions.. " Commun. Math. Phys. 153, 297-320 (1993).
19. Simon, B., " Resonances in n -body quantum systems with dilatation analytic potentials., " Ann. Math. 97, 247-274 (1973).
20. Strauss, W. A., " Existence of solitary waves in higher dimensions., " Commun. Math. Phys. 55, 149-162 (1977).
21. Soffer, A., and Weinstein, M. I., " Multichannel nonlinear scattering for nonintegrable equations., " Commun. Math. Phys. 133, 119-146 (1990).
22. Soffer, A., and Weinstein, M. I., " Multichannel nonlinear scattering for nonintegrable equations. II. The case of anisotropic potentials and data., " J. Differ. Equations 98 376-390 (1992).
23. Soffer, A., and Weinstien, M. I., " Selection of the ground state for nonlinear Schrödinger equations., " Rev. Math. Phys. 16, 977-1071 (2004).
24. Soffer, A., and Weinstein, M. I., " Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations., " Invent. Math. 136, 9-74 (1999).
25. Tsai, T.-P., and Yau, H.-T., " Asymptotic dynamics of nonlinear Schrödinger equations., " Commun. Pure Appl. Math. LV, 0153-0216 (2002a).
26. Tsai, T.-P., and Yau, H.-T., " Stable directions for excited states of nonlinear Schrödinger equations., " Commun. Partial Differ. Equ. 27 11-12 (2002b).
27. Tsai, T.-P., and Yau, H.-T., " Relaxation of excited states in nonlinear Schrödinger equations., " Int. Math. Res. Notices 2002, 1629-1673 (2002c).
28. Weinstein, M. I., " Lyapunov stability of ground states of nonlinear dispersive evolution equations., " Commun. Pure Appl. Math. 39, 51-67 (1986).
29. Fröhlich and Spencer, (private communication) (2005).
30. Tsai, T.-P. (private communication) (2006).