Invariant Gibbs measure for a Schrodinger equation with exponential nonlinearity

TL;DR

This paper studies the invariance of the Gibbs measure for a fractional exponential nonlinear Schrödinger equation on compact manifolds, establishing existence, invariance, and global solutions under various conditions.

Contribution

It constructs and analyzes the Gibbs measure for expNLS, proving invariance and global well-posedness results in different regimes of dispersion and nonlinearity.

Findings

Gibbs measure well-defined in defocusing case for all parameters

Partition function infinite in focusing case regardless of parameters

Solutions are almost surely global in large dispersion regime

Abstract

We investigate the invariance of the Gibbs measure for the fractional Schrodinger equation of exponential type (expNLS) $i\partial_t u + (-\Delta)^{\frac{\alpha}2} u = 2\gamma\beta e^{\beta|u|^2}u$ on $d$-dimensional compact Riemannian manifolds $\mathcal{M}$, for a dispersion parameter $\alpha>d$, some coupling constant $\beta>0$, and $\gamma\neq 0$. (i) We first study the construction of the Gibbs measure for (expNLS). We prove that in the defocusing case $\gamma>0$, the measure is well-defined in the whole regime $\alpha>d$ and $\beta>0$ (Theorem 1.1 (i)), while in the focusing case $\gamma<0$ its partition function is always infinite for any $\alpha>d$ and $\beta>0$, even with a mass cut-off of arbitrary small size (Theorem 1.1 (ii)). (ii) We then study the dynamics (expNLS) with random initial data of low regularity. We first use a compactness argument to prove weak invariance of the Gibbs measure in the whole regime $\alpha>d$ and $0<\beta < \beta^\star_\alpha$ for some natural parameter $0<\beta^\star_\alpha\sim (\alpha-d)$ (Theorem 1.3 (i)). In the large dispersion regime $\alpha>2d$, we can improve this result by constructing a local deterministic flow for (expNLS) for any $\beta>0$. Using the Gibbs measure, we prove that solutions are almost surely global for $0<\beta \ll\beta^\star_\alpha$, and that the Gibbs measure is invariant (Theorem 1.3 (ii)). (iii) Finally, in the particular case $d=1$ and $\mathcal{M}=\mathbb{T}$, we are able to exploit some probabilistic multilinear smoothing effects to build a probabilistic flow for (expNLS) for $1+\frac{\sqrt{2}}2<\alpha \leq 2$, locally for arbitrary $\beta>0$ and globally for $0<\beta \ll \beta^\star_\alpha$ (Theorem 1.5).