Multichannel nonlinear scattering theory for nonintegrable equations

  • A. Soffer
  • M. I. Weinstein
Conference paper

DOI: 10.1007/BFb0035679

Part of the Lecture Notes in Physics book series (LNP, volume 342)
Cite this paper as:
Soffer A., Weinstein M.I. (1989) Multichannel nonlinear scattering theory for nonintegrable equations. In: Balabane M., Lochak P., Sulem C. (eds) Integrable Systems and Applications. Lecture Notes in Physics, vol 342. Springer, Berlin, Heidelberg

Abstract

We consider a class of nonlinear equations with localized and dispersive solutions; we show that for a ball in some Banach space of initial conditions, the asymptotic behavior (as t → ±∞) of such states is given by a linear combination of a periodic (in time), localized (in space) solution (nonlinear bound state) of the equation and a purely dispersive part (with free dispersion). We also show that given data near a nonlinear bound state of the system, there is a nonlinear bound state of nearby energy and phase, such that the difference between the solution (adjusted by a phase) and the latter disperses to zero. It turns out that in general the time-period (and energy) of the localized part is different for t → ±∞from that for t → −∞.Moreover, the solution acquires an extra constant phase eiγ±.

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Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • A. Soffer
    • 1
  • M. I. Weinstein
    • 2
  1. 1.Department of MathematicsPrinceton UniversityPrinceton
  2. 2.Department of MathematicsUniversity of MichiganAnn Arbor

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