A Microscopic Derivation of Gibbs Measures for the 1D Focusing Quintic Nonlinear Schrödinger Equation

In this work, we obtain a microscopic derivation of Gibbs measures for the focusing quintic nonlinear Schrödinger equation on � 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 [A. Rout and V. Sohinger, Comm. Partial Differential Equations, 48 (2023), pp. 1008–1055]. 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 Fröhlich, Knowles, Schlein, and the second author [Comm. Math. Phys., 356 (2017), pp. 883–980]. We prove results in both the time-independent and the time-dependent setting.