Invariance of the Gibbs Measure for the Schrodinger-Benjamin-Ono System

TL;DR

This paper proves the invariance of the Gibbs measure for the periodic Schrödinger-Benjamin-Ono system under certain conditions by establishing new well-posedness results and constructing the measure in a specific Sobolev space.

Contribution

It introduces a new local well-posedness framework and constructs the Gibbs measure for the system in a sub-L^2 setting, extending understanding of its invariance.

Findings

Gibbs measure invariance established for |b3| 1.

Ill-posedness results for s < 1/2 when |b3| 1.

Ill-posedness for all s when |b3| = 1.

Abstract

We prove the invariance of the Gibbs measure for the periodic Schrodinger-Benjamin-Ono system (when the coupling parameter |\gamma| \ne 0, 1) by establishing a new local well-posedness in a modified Sobolev space and constructing the Gibbs measure (which is in the sub-L^2 setting for the Benjamin-Ono part.) We also show the ill-posedness result in H^s(\mathbb{T}) \times H^{s-{1/2}}(\mathbb{T}) for s < {1/2} when |\gamma| \ne 0, 1 and for any s \in \mathbb{R} when |\gamma| =1.