Mathematics – Analysis of PDEs
Scientific paper
2012-03-19
Mathematics
Analysis of PDEs
99 pages. Added more references in new version
Scientific paper
The soliton resolution conjecture for the focusing nonlinear Schrodinger equation (NLS) is the vaguely worded claim that a global solution of the NLS, for generic initial data, will eventually resolve into a radiation component that disperses like a linear solution, plus a localized component that behaves like a soliton or multi-soliton solution. Considered to be one of the fundamental open problems in the area of nonlinear dispersive equations, this conjecture has eluded a proof or even a precise formulation till date. This paper proves a "statistical version" of this conjecture at mass-subcritical nonlinearity, in the following sense. The uniform probability distribution on the set of all functions with a given mass and energy, if such a thing existed, would be a natural invariant measure for the NLS flow and would reflect the long-term behavior for "generic initial data" with that mass and energy. Unfortunately, such a probability measure does not exist. We circumvent this problem by constructing a sequence of discrete measures that, in principle, approximate this fictitious probability distribution as the grid size goes to zero. We then show that a continuum limit of this sequence of probability measures does exist in a certain sense, and in agreement with the soliton resolution conjecture, the limit measure concentrates on the unique ground state soliton. This is proved by first proving a similar result for the discrete NLS. The above result, following in the footsteps of a program of studying the long-term behavior of nonlinear dispersive equations through their natural invariant measures initiated by Lebowitz, Rose and Speer, and carried forward by Bourgain, McKean, Tzvetkov, Oh and others, is proved using a combination of techniques from large deviations, PDE, harmonic analysis and bare hands probability theory. It is valid in any dimension.
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