1203.4027 (Sourav Chatterjee)
Sourav Chatterjee
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.
View original:
http://arxiv.org/abs/1203.4027
No comments:
Post a Comment