Invariant measures and the soliton resolution conjecture
New challenges in PDE: Deterministic dynamics and randomness in high and infinite dimensional systems October 19, 2015 - October 30, 2015
Location: MSRI: Simons Auditorium
discrete NLS on a torus
Birkhoff's ergodicity theorem
existence of global solutions
58J51 - Relations between spectral theory and ergodic theory, e.g. quantum unique ergodicity
28D10 - One-parameter continuous families of measure-preserving transformations
28C10 - Set functions and measures on topological groups or semigroups, Haar measures, invariant measures [See also 22Axx, 43A05]
37P30 - Height functions; Green functions; invariant measures [See also 11G50, 14G40]
35R15 - Partial differential equations on infinite-dimensional (e.g. function) spaces (= PDE in infinitely many variables) [See also 46Gxx, 58D25]
I will talk about the micro-canonical invariant measure for the discrete nonlinear Schrödinger equation on a torus in the mass-subcritical regime, and prove that a random function drawn from this measure is close to the ground state soliton with high probability. This proves that “almost all” ergodic components of this flow have the property of convergence to a soliton in the long run, which is a statistical variant of what is sometimes called the soliton resolution conjecture
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