Selection of the ground state for nonlinear schrödinger equations

A. Soffer, M. I. Weinstein

Research output: Contribution to journalReview articlepeer-review

97 Scopus citations

Abstract

We prove for a class of nonlinear Schrödinger systems (NLS) having two nonlinear bound states that the (generic) large time behavior is characterized by decay of the excited state, asymptotic approach to the nonlinear ground state and dispersive radiation. Our analysis elucidates the mechanism through which initial conditions which are very near the excited state branch evolve into a (nonlinear) ground state, a phenomenon known as ground state selection. Key steps in the analysis are the introduction of a particular linearization and the derivation of a normal form which reflects the dynamics on all time scales and yields, in particular, nonlinear master equations. Then, a novel multiple time scale dynamic stability theory is developed. Consequently, we give a detailed description of the asymptotic behavior of the two bound state NLS for all small initial data. The methods are general and can be extended to treat NLS with more than two bound states and more general nonlinearities including those of Hartree-Fock type.

Original languageEnglish (US)
Pages (from-to)977-1071
Number of pages95
JournalReviews in Mathematical Physics
Volume16
Issue number8
DOIs
StatePublished - Sep 1 2004

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Keywords

  • Asymptotic stability
  • Gross-pitaevskii equation
  • Metastability
  • NLS
  • Nonlinear scattering
  • Soliton dynamics

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