A microscopic derivation of Gibbs measures for the 1D focusing quintic nonlinear Schr\"{o}dinger equation

TL;DR

This paper derives Gibbs measures for the 1D focusing quintic nonlinear Schrödinger equation from many-body quantum states, extending previous cubic case work to the more complex three-body interaction scenario.

Contribution

It provides the first microscopic derivation of Gibbs measures for the quintic NLS, including time-dependent correlation functions, in the three-body interaction regime.

Findings

First microscopic derivation of Gibbs measures for 1D focusing quintic NLS.

Extension of methods to three-body interactions from previous two-body work.

Establishment of results in both time-independent and time-dependent settings.

Abstract

In this work, we obtain a microscopic derivation of Gibbs measures for the focusing quintic nonlinear Schr\"{o}dinger equation (NLS) on $\mathbb{T}$ from many-body quantum Gibbs states. On the quantum many-body level, the quintic nonlinearity corresponds to a three-body interaction. This is a continuation of our previous work. In the aforementioned work, we studied the cubic problem, which corresponds to a two-body interaction on the quantum many-body level. In our setup, we truncate the mass of the classical free field in the classical setting and the rescaled particle number in the quantum setting. Our methods are based on a perturbative expansion previously developed in the work of Fr\"{o}hlich, Knowles, Schlein, and the second author. We prove results both in the time-independent and time-dependent setting. This is the first such known result in the three-body regime. Furthermore, this gives the first microscopic derivation of time-dependent correlation functions for Gibbs measures corresponding to the quintic NLS, as studied in the work of Bourgain.