×

Multichannel nonlinear scattering for nonintegrable equations. (English) Zbl 0721.35082

The authors consider a class of nonlinear (time dependent) Schrödinger equations with localized and dispersive solutions. They obtain a class of initial conditions, for which the asymptotic behavior of solutions is given by a linear combination of nonlinear bound state and a purely dispersive part. They obtain also asymptotic stability result.
Reviewer: D.Robert (Nantes)

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35B40 Asymptotic behavior of solutions to PDEs
35B35 Stability in context of PDEs
35P25 Scattering theory for PDEs
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] [Ag] Agmon, S.: Lectures on Exponential Decay of Solutions of Second Order Elliptic Equations: Bounds on Eigenfunctions ofN-Body Schrödinger Operators. Princeton, NJ: Princeton University Press 1982 · Zbl 0503.35001
[2] [Ben] Benjamin, T. B.: The stability of solitary waves. Proc. R. Soc. Lond.A328, 153 (1972)
[3] [Ber] Berry, M. V.: Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. Lond.A392, 45 (1984) · Zbl 1113.81306
[4] [Be-Li] Berestycki, H., Lions, P.-L.: Nonlinear scalar field equations I-Existence of a ground state. Arch. Rat. Mech. Anal.82, 313–345 (1983) · Zbl 0533.35029
[5] [Ca-Li] Cazenave, T., Lions, P.-L.: Orbital stability of standing waves for some nonlinear Schrödinger equations. Commun. Math. Phys.85, 549–561 (1982) · Zbl 0513.35007
[6] [C-K] Cohen, A., Kappeler, T.: preprint
[7] [C-R] Crandall, M., Rabinowitz, P.: Bifurcation of simple eigenvalues and linearized stability. Arch. Rat. Mech. Anal.52, 161–181 (1973) · Zbl 0275.47044
[8] [C-W] Cazenave, T., Weissler, F. B.: The Cauchy problem for the nonlinear Schrödinger equation inH 1, 1989 preprint
[9] [En] Enss, V.: Quantum Scattering Theory of Two and Three Body Systems with Potentials of Short and Long Range. In: Schrödinger Operators, Graffi, S. (ed.). Lecture Notes in Mathematics. Berlin, Heidelberg, New York: Springer 1985 · Zbl 0585.35023
[10] [G-G-K-M] Gardner, C. S., Greene, J. M., Kruskal, M. D., Miura, R. M.: Method for solving the Korteweg-de Vries equation. Phys. Rev. Lett.19, 1095–1097 (1967) · Zbl 1103.35360
[11] [G-S-S] Grillakis, M., Shatah, J., Strauss, W.: Stability theory of solitary waves in the presence of symmetry, I. J. Func. Anal.74, 760–797 (1988)
[12] [G-V] Ginibre, J., Velo, G.: On a class of nonlinear Schrödinger equations I, II. J. Func. Anal.32, 1–71 (1979) · Zbl 0396.35028
[13] [Gl] Glassey, R. T.: On the blowing up of solutions to the Cauchy problem for the nonlinear Schrödinger equation. J. Math. Phys.18, 1794–1797 (1977) · Zbl 0372.35009
[14] [H-N-T] Hayashi, N., Nakamitsu, K., Tsutsumi, M.: On solutions of the initial value problem for the nonlinear Schrödinger equations. J. Func. Anal.71, 218–245 (1987) · Zbl 0657.35033
[15] [J-K] Jensen, A., Kato, T.: Spectral properties of Schrödinger operators and time decay of the wave functions. Duke Math. J.46, 583–611 (1979) · Zbl 0448.35080
[16] [K] Kato, T.: On nonlinear Schrödinger equations. Ann. Inst. Henri Poincaré, Physique Théorique46, 113–129 (1987)
[17] [K-A] Kodama, Y., Ablowitz, M. J.: Perturbations of solitons and solitary waves. Stud. Appl. Math.64, 225–245 (1981) · Zbl 0486.76029
[18] [K-M] Keener, J. P., McLaughlin, D. W.: Solitons under perturbations. Phys. Rev.A16, 777–790 (1977)
[19] [Lax] Lax, P. D.: Integrals of nonlinear equations of evolution and solitary waves. Commun. Pure Appl. Math.21, 467–490 (1968) · Zbl 0162.41103
[20] [Li] Bishop, A., Campbell, D., Nicolaenko, B.: Nonlinear Problems: Present and Future. North Holland Math. Studies
[21] [Mu] Murata, M.: Rate of decay of local energy and spectral properties of elliptic operators. Jpn. J. Math.6, 77–127 (1980) · Zbl 0447.35013
[22] [Ne] Newell, A. C.: Near-integrable systems, nonlinear tunneling and solitons in slowly changing media. Nonlinear Evolution Equations Solvable by the Inverse Spectral Trans form. Calogero, F. (ed.), pp. 127–179. London: Pitman 1978
[23] [Nir] Nirenberg, L.: Topics in Nonlinear Functional Analysis. Courant Institute Lecture Notes, 1974
[24] [Ra] Rauch, J.: Local decay of scattering solutions to Schrödinger’s equation. Commun. Math. Phys.61, 149–168 (1978) · Zbl 0381.35023
[25] [R-S] Reed, M. Simon, B.: Methods of Mathematical Physics, II, III, IV. New York: Academic Press
[26] [Ro-We] Rose, H. A., Weinstein, M. I.: On the bound states of the nonlinear Schrödinger equation with a linear potential. PhysicaD30, 207–218 (1988) · Zbl 0694.35202
[27] [Str1] Strauss, W. A.: Dispersion of low energy waves for two conservative equations. Arch. Rat. Mech. Anal.55, 86–92 (1974) · Zbl 0289.35048
[28] [Str2] Strauss, W. A.: Existence of solitary waves in higher dimensions. Commun. Math. Phys.55, 149–162 (1977) · Zbl 0356.35028
[29] [Str3] Strauss, W. A.: Nonlinear scattering theory at low energy. J. Func. Anal.41, 110–133 (1981) · Zbl 0466.47006
[30] [Sh-Str] Shatah, J., Strauss, W.: Instability of nonlinear bound states. Commun. Math. Phys.100, 173–190 (1985) · Zbl 0603.35007
[31] [Sig-Sof] Sigal, I. M., Soffer, A.: TheN-particle scattering problem: Asymptotic completeness for short range systems. Ann. Math.126, 35–108 (1987) · Zbl 0646.47009
[32] [Sof-We] Soffer, A., Weinstein, M. I.: Multichannel nonlinear scattering theory for nonintegrable equations. Proceedings of the Oléron Conference on Integrable Systems, 1988. Lecture Notes in Physics No. 342. Berlin, Heidelberg, New York: Springer
[33] [We1] Weinstein, M. I.: Nonlinear Schrödinger equations and sharp interpolation estimates. Commun. Math. Phys.87, 567–576 (1983) · Zbl 0527.35023
[34] [We2] Weinstein, M. I.: Modulational stability of ground states of nonlinear Schrödinger equations. SIAM J. Math. Anal.16, 472–491 (1985) · Zbl 0583.35028
[35] [We3] Weinstein, M. I.: Lyapunov stability of ground states of nonlinear dispersive evolution equations. Commun. Pure Appl. Math.39, 51–68 (1986) · Zbl 0594.35005
[36] [Z-S] Zakharov, V. E., Shabat, A. B.: Exact theory of two dimensional self focusing and one dimensional self modulation of waves in nonlinear media. Sov. Phys. J.E.T.P.34, 62–69 (1972)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.