arXiv subjects

Caroline Brett

Publications and source records attributed to Caroline Brett.

3 recordsLinked to original sources

arXiv2019

Statistical Mechanics of the Periodic Benjamin-Ono Equation

The periodic Benjamin-Ono equation is an autonomous Hamiltonian system with a Gibbs measure on $L^2({\mathbb T})$. 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 $n\rightarrow\infty$, 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.

math.AP↗

arXiv2014

Hill's Spectral Curves and the Invariant Measure of the Periodic KdV Equation

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 $ν_N^β$ of the periodic KdV equation. This $ν_N^β$ is the modified canonical ensemble, as given by Bourgain ({Comm. Math. Phys.} {166} (1994), 1--26), and $ν_N^β$ satisfies a logarithmic Sobolev inequality. Associated concentration inequalities control the fluctuations of the periodic eigenvalues $(λ_n)$. For $β, N>0$ small, there exists a set of positive $ν_N^β$ measure such that $(\pm \sqrt{2(λ_{2n}+λ_{2n-1})})_{n=0}^\infty$ gives a sampling sequence for Paley--Wiener space $PW(π)$ and the reproducing kernels give a Riesz basis. Let $(μ_j)_{j=1}^\infty$ be the tied spectrum; then $(2\sqrt{μ_j}-j)$ belongs to a Hilbert cube in $\ell^2$ and is distributed according to a measure that satisfies Gaussian concentration for Lipschitz functions. The sampling sequence $(\sqrt{μ_j})_{j=1}^\infty$ arises from a divisor on the spectral curve, which is hyperelliptic of infinite genus. The linear statistics $\sum_j g(\sqrt{λ_{2j}})$ with test function $g\in PW(π)$ satisfy Gaussian concentration inequalities.

math.SP↗

arXiv2014

Logarithmic Sobolev inequalities and spectral concentration for the cubic Schrödinger equation

The nonlinear Schrödinger equation NLSE(p, β), -iu_t=-u_{xx}+β| u|^{p-2} u=0, arises from a Hamiltonian on infinite-dimensional phase space \Lp^2(\mT). For p\leq 6, Bourgain (Comm. Math. Phys. 166 (1994), 1--26) has shown that there exists a Gibbs measure μ^β_N on balls Ω_N= {ϕ\in \Lp^2(\mT) : | ϕ|^2_{\Lp^2} \leq 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\leq p\leq 4 on Ω_N for all N>0; also μ^β satisfies a restricted LSI for 4\leq p\leq 6 on compact subsets of Ω_N determined by Hölder norms. Hence for p=4, the spectral data of the periodic Dirac operator in \Lp^2(\mT; \mC^2) with random potential ϕsubject 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.

math.SP↗