Selection of the ground state for nonlinear Schrödinger Equations

Michael Weinstein
Bell Laboratories
Fundamental Mathematics Research Department

We consider the nonlinear Schrödinger equation with a spatially localized
potential well. If the potential well supports two bound states, then for
generic initial conditions the solution's energy asymptotically (t
→ ∞) partitions between a nonlinear ground
state and dispersive radiation. In particular, the excited state decays. We give
a detailed analytical description of this phenomenon on small, large but finite,
and infinite time scales. Two examples where this phenomenon arises are in (1)
the propagation of "gap solitons" in periodic structures with spatially
localized defects and (2) the evolution of solitary waves for the Gross-Pitaevskii
equation.


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