Gaussian ensembles for the non-linear Schrödinger and KdV equations.

Blower, Gordon (2001) Gaussian ensembles for the non-linear Schrödinger and KdV equations. Stochastics and Stochastics Reports, 71 (3-4). pp. 177-200.

Abstract

Let be the soliton solution to the nonlinear Schrdinger equation on the line. Following the approach of Lebowitz et al. (J. Statist. Phys. 54, 17-56 (1989)) to the periodic case, a family of Gaussian ensembles is introduced. This approximates the Gibbs measure in the sense that it is concentrated on locally bounded functions which are locally uniformly close to the soliton solution. The measure may be normalized when the inverse temperature is sufficiently small. The covariance matrix of the Gaussian process satisfies the Schrdinger equation obtained by linearizing the original equation about the soliton solution. Further, the Gaussian process is stationary with respect to time-shift and spatial translation, in Levitan's sense. Gaussian ensembles for the modified KdV equation are also introduced