The nonlinear Schrödinger equation NLSE(p, β), −iut = −uxx + β|u| p−2 u = 0, arises from a Hamiltonian on infinite-dimensional phase space L 2 (T). For p ≤ 6, Bourgain (Comm. Math. Phys. 166 (1994), 1-26) has shown that there exists a Gibbs measure µL 2 ≤ N } in phase space such that the Cauchy problem for NLSE(p, β) is well posed on the support of µ β N , and that µ β N is invariant under the flow. This paper shows that µ β N satisfies a logarithmic Sobolev inequality for the focussing case β < 0 and 2 ≤ p ≤ 4 on Ω N for all N > 0; also µ β satisfies a restricted LSI for 4 ≤ p ≤ 6 on compact subsets of Ω N determined by Hölder norms. Hence for p = 4, the spectral data of the periodic Dirac operator in L 2 (T; C 2 ) with random potential φ subject to µ β N are concentrated near to their mean values. The paper concludes with a similar result for the spectral data of Hill's equation when the potential is random and subject to the Gibbs measure of KdV.

The periodic Benjamin-Ono equation is an autonomous Hamiltonian system with a Gibbs measure on L 2 (T). The paper shows that the Gibbs measures on bounded balls of L 2 satisfy some logarithmic Sobolev inequalities. The space of n-soliton solutions of the periodic Benjamin-Ono equation, as discovered by Case, is a Hamiltonian system with an invariant Gibbs measure. As n → ∞, these Gibbs measures exhibit a concentration of measure phenomenon. Case introduced soliton solutions that are parameterised by atomic measures in the complex plane. The limiting distributions of these measures gives the density of a compressible gas that satisfies the isentropic Euler equations.Date: 1st February 2019.

This paper analyses the periodic spectrum of Schrödinger's equation −f ′′ +qf = λf when the potential is real, periodic, random and subject to the invariant measure ν and is distributed according to a measure that satisfies Gaussian concentration for Lipschitz functions. The sampling sequence (arises from a divisor on the spectral curve, which is hyperelliptic of infinite genus. The linear statistics j g( λ 2j ) with test function g ∈ P W (π) satisfy Gaussian concentration inequalities.