Solitons and Gibbs measures for nonlinear Schroedinger equations

TL;DR

This paper reviews recent advances in Gibbs measures for nonlinear Schrödinger equations, highlighting phase transitions and stability of solitary waves, with implications for understanding the typical behavior of solutions.

Contribution

It provides a comprehensive review of Gibbs measures in NLS, including new results on phase transitions and soliton stability in discrete 3D models.

Findings

Identification of phase transition to soliton-like behavior in 3D discrete NLS

Insights into stability and typicality of solitary wave structures

Implications for the mathematical understanding of NLS dynamics

Abstract

We review some recent results concerning Gibbs measures for nonlinear Schroedinger equations (NLS), with implications for the theory of the NLS, including stability and typicality of solitary wave structures. In particular, we discuss the Gibbs measures of the discrete NLS in three dimensions, where there is a striking phase transition to soliton-like behavior.