Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
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TY - JOUR
T1 - Gaussian ensembles for the non-linear Schrödinger and KdV equations.
AU - Blower, Gordon
PY - 2001
Y1 - 2001
N2 - 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
AB - 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
KW - Gibbs measure
KW - Stationary stochastic process
KW - Nonlinear Schrodinger equation
U2 - 10.1080/17442500108834265
DO - 10.1080/17442500108834265
M3 - Journal article
VL - 71
SP - 177
EP - 200
JO - Stochastics and Stochastics Reports
JF - Stochastics and Stochastics Reports
IS - 3-4
ER -