Well-posedness and invariant measures for complex valued modified KdV equation

TL;DR

This paper establishes the well-posedness and constructs invariant measures for the complex-valued periodic modified KdV equation, advancing understanding of its dynamics in Sobolev spaces despite the added complexity of complex values.

Contribution

It introduces a novel approach to construct invariant measures for the complex-valued mKdV, extending previous real-valued analyses to the complex setting.

Findings

Invariant measures supported on Sobolev spaces with increasing regularity

Analysis of complex-valued functions increases the difficulty of invariance proofs

Method adapts ideas from Zhidkov for real-valued cases to complex-valued equations

Abstract

We consider the one dimensional periodic complex valued mKdV, which corresponds to the first equation above cubic NLS in the associated integrable hierarchy. Our main result is the construction of a sequence of invariant measures supported on Sobolev spaces with increasing regularity. The fact that we work with complex valued functions makes the analysis of the invariance much harder compared to the real valued case, that can be handled instead following the ideas used by Zhidkov [73].