TY - JOUR

T1 - Logarithmic Sobolev inequalities and spectral concentration for the cubic Shrödinger equation

AU - Blower, Gordon

AU - Brett, Caroline

AU - Doust, Ian

N1 - Submitted for publication, and posted on ArXiv 1308:3649 The final, definitive version of this article has been published in the Journal, Stochastics, 86 (6), 2014, © Informa Plc

PY - 2014

Y1 - 2014

N2 - The nonlinear Schr\"odinger equation $\NLSE(p, \beta)$,$-iu_t=-u_{xx}+\beta \vert u\vert^{p-2} u=0$,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 $\mu^{\beta}_N$ on balls $\Omega_N=\{ \phi\in \Lp^2(\mT) \,:\, \Vert \phi \Vert^2_{\Lp^2} \leq N\}$in phase space such that the Cauchy problem for $\NLSE(p,\beta)$ is well posed on the support of $\mu^{\beta}_N$, and that $\mu^{\beta}_N$ is invariant underthe flow.This paper shows that $\mu^{\beta}_N$ satisfies a logarithmic Sobolev inequality for the focussing case $\beta <0$ and $2\leq p\leq 4$on $\Omega_N$ for all $N>0$; also $\mu^{\beta}$ satisfies a restricted LSI for $4\leq p\leq 6$ on compact subsets of $\Omega_N$ determined by H\"oldernorms. Hence for $p=4$, the spectral data of the periodic Dirac operatorin $\Lp^2(\mT; \mC^2)$ with random potential $\phi$ subject to $\mu^{\beta}_N$ 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.

AB - The nonlinear Schr\"odinger equation $\NLSE(p, \beta)$,$-iu_t=-u_{xx}+\beta \vert u\vert^{p-2} u=0$,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 $\mu^{\beta}_N$ on balls $\Omega_N=\{ \phi\in \Lp^2(\mT) \,:\, \Vert \phi \Vert^2_{\Lp^2} \leq N\}$in phase space such that the Cauchy problem for $\NLSE(p,\beta)$ is well posed on the support of $\mu^{\beta}_N$, and that $\mu^{\beta}_N$ is invariant underthe flow.This paper shows that $\mu^{\beta}_N$ satisfies a logarithmic Sobolev inequality for the focussing case $\beta <0$ and $2\leq p\leq 4$on $\Omega_N$ for all $N>0$; also $\mu^{\beta}$ satisfies a restricted LSI for $4\leq p\leq 6$ on compact subsets of $\Omega_N$ determined by H\"oldernorms. Hence for $p=4$, the spectral data of the periodic Dirac operatorin $\Lp^2(\mT; \mC^2)$ with random potential $\phi$ subject to $\mu^{\beta}_N$ 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.

KW - nonlinear Schroedinger equation

KW - Gibbs measure

KW - Hill's equation

U2 - 10.1080/17442508.2014.882924

DO - 10.1080/17442508.2014.882924

M3 - Journal article

VL - 86

SP - 870

EP - 881

JO - Stochastics

JF - Stochastics

SN - 1744-2508

IS - 6

ER -