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Hasimoto frames and the Gibbs measure of periodic nonlinear Schr\"odinger Equation

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arxiv 2205.12868 v1 pith:XNNOSXDZ submitted 2022-05-25 math.AP math.PR

Gordon Blower , Azadeh Khaleghi , Moe Kuchemann-Scales This is my paper · ORCID
classification math.APmath.PR
keywords measuregibbsframesequationhasimotomeasuresmovingnonlinear
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abstract

The paper interprets the cubic nonlinear Schr\"odinger equation as a Hamiltonian system with infinite dimensional phase space. There is 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, NLSE gives a Lax pair of coupled ODE 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.

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