Mathematical Modelling of Natural Phenomena

Table of Contents - 2012 - Volume 7, Issue 02 (Solitary waves)  

Research Article

Solitons and Gibbs Measures for Nonlinear Schrödinger Equations

K. Kirkpatrick 

University of Illinois at Urbana-Champaign, Department of Mathematics, Urbana, IL, 61801, USA

Abstract

We review some recent results concerning Gibbs measures for nonlinear Schrödinger 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.

(Online publication February 29 2012)

Key Words:

  • NLS equation;
  • statistical mechanics;
  • invariant Gibbs measures;
  • exact solvability

Mathematics Subject Classification:

  • 35Q55;
  • 81V70;
  • 81Q80

Correspondence:

Corresponding author. E-mail: kkirkpat@illinois.edu