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

Research output: Contribution to journalJournal articlepeer-review

<mark>Journal publication date</mark>2001
<mark>Journal</mark>Stochastics and Stochastics Reports
Issue number3-4
Number of pages24
Pages (from-to)177-200
Publication StatusPublished
<mark>Original language</mark>English


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