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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 - Hasimoto frames and the Gibbs measure of the periodic nonlinear Schrödinger equation
AU - Blower, Gordon
AU - Khaleghi, Azadeh
AU - Kuchemann-Scales, Moe
PY - 2024/2/1
Y1 - 2024/2/1
N2 - The paper interprets the cubic nonlinear Schrödinger equation as a Hamiltonian system with infinite dimensional phase space. There exists a Gibbs measure which is invariant under the flow associated with the canonical equations of motion. The logarithmic Sobolev and concentration of measure inequalities hold for the Gibbs measures, and here are extended to the k-point correlation function and distributions of related empirical measures. By Hasimoto’s theorem, the nonlinear Schrödinger equation gives a Lax pair of coupled ordinary differential equations for which the solutions give a system of moving frames. The paper studies the evolution of the measure induced on the moving frames by the Gibbs measure; the results are illustrated by numerical simulations. The paper contains quantitative estimates with well-controlled constants on the rate of convergence of the empirical distribution in Wasserstein metric.
AB - The paper interprets the cubic nonlinear Schrödinger equation as a Hamiltonian system with infinite dimensional phase space. There exists a Gibbs measure which is invariant under the flow associated with the canonical equations of motion. The logarithmic Sobolev and concentration of measure inequalities hold for the Gibbs measures, and here are extended to the k-point correlation function and distributions of related empirical measures. By Hasimoto’s theorem, the nonlinear Schrödinger equation gives a Lax pair of coupled ordinary differential equations for which the solutions give a system of moving frames. The paper studies the evolution of the measure induced on the moving frames by the Gibbs measure; the results are illustrated by numerical simulations. The paper contains quantitative estimates with well-controlled constants on the rate of convergence of the empirical distribution in Wasserstein metric.
U2 - 10.1063/5.0169792
DO - 10.1063/5.0169792
M3 - Journal article
VL - 65
JO - Journal of Mathematical Physics
JF - Journal of Mathematical Physics
SN - 0022-2488
IS - 2
M1 - 022705
ER -