Vol. 1, No. 3, 2008

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Dynamics of nonlinear Schrödinger/Gross–Pitaevskii equations: mass transfer in systems with solitons and degenerate neutral modes

Zhou Gang and Michael I. Weinstein

Vol. 1 (2008), No. 3, 267–322

Nonlinear Schrödinger/Gross–Pitaevskii equations play a central role in the understanding of nonlinear optical and macroscopic quantum systems. The large time dynamics of such systems is governed by interactions of the nonlinear ground state manifold, discrete neutral modes (“excited states”) and dispersive radiation. Systems with symmetry, in spatial dimensions larger than one, typically have degenerate neutral modes. Thus, we study the large time dynamics of systems with degenerate neutral modes. This requires a new normal form (nonlinear matrix Fermi Golden Rule) governing the system’s large time asymptotic relaxation to the ground state (soliton) manifold.

soliton, nonlinear bound state, nonlinear scattering, asymptotic stability, dispersive partial differential equation
Mathematical Subject Classification 2000
Primary: 35Q51, 37K40, 37K45
Received: 9 January 2008
Revised: 14 July 2008
Accepted: 22 October 2008
Published: 18 December 2008
Zhou Gang
Department of Mathematics
Princeton University
Princeton, NJ
United States
Michael I. Weinstein
Department of Applied Physics and Applied Mathematics
Columbia University
New York, NY 10027
United States