- 03-398 A. Soffer and M.I. Weinstein
- Selection of the Ground State for Nonlinear Schroedinger Equations
(950K, Zipped Postscript)
Sep 1, 03
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Abstract. We prove for a class of nonlinear Schroedinger 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.