The phase space of the focused cubic Schroedinger equation: A numerical study

Abstract

Yuri 0. Burlakov me Goverment reserves for itself and others acting on its behalf a royalty free, nonexclusive, irrevocable, worldwide license for govermental ~xlrposes t

Probabilistic Methods for Discrete Nonlinear Schrödinger Equations

“…Now, if f ∈ V a 0 and f ∈ W c 0 , then a(f ) > a 0 and b(f )−a(f ) > c 0 . Therefore by (24), (25) and (26), it follows that if n > C, A combination of the above inequality, (23), (9) and Lemma 3.6 proves the conclusion of Theorem 2.1 when βB 2 > θ c .…”

“…Then by (31), (24), (25), (23), (9) and the observation that r > max{4/5 + , 2p}, we get that if n > C,…”

Solitons and Gibbs Measures for Nonlinear Schrödinger Equations

“…By contrast, simulations in [10] suggested that there is no phase transition, and then Rider, following on the works of McKean and Vaninsky [41,42,43], confirmed this for the 1D infinite-volume focusing NLS by proving that the thermodynamic limit is trivial [52,53]. Bourgain also studied invariant measures of the 1D infinite-volume defocusing NLS [5] and of the 2D defocusing NLS [6] (see also the review article [7]), as did Tzvetkov [59].…”

On the ∞‐volume limit of the focusing cubic Schrödinger equation

“…The outcome suggested a possible phase transition: the ensemble preferring radiation/solitons at low/high values of D or T . Numerical work of [1] ran contrary to this. McKean then put forward a proof that the full thermodynamic limit does not exist ( [6]); that is, depending on how the circle is taken to the whole line, one sees an infinite number of limiting processes.…”