Periodic fourth-order cubic NLS: Local well-posedness and Non-squeezing property

TL;DR

This paper establishes local well-posedness for the periodic cubic fourth-order nonlinear Schrödinger equation in low regularity spaces and proves the non-squeezing property of its flow, contributing to the understanding of its mathematical structure.

Contribution

It provides the first proof of local well-posedness in low regularity Sobolev spaces and demonstrates the non-squeezing property for the 4NLS flow on the torus.

Findings

Well-posedness in $H^s$ for $-1/3 \,\leq s < 0$

Non-squeezing property of the flow map in $L^2(\mathbb{T})$

Extension of techniques from previous NLS studies

Abstract

In this paper, we consider the cubic fourth-order nonlinear Schr\"odinger equation (4NLS) under the periodic boundary condition. We prove two results. One is the local well-posedness in $H^s$ with $-1/3 \le s < 0$ for the Cauchy problem of the Wick ordered 4NLS. The other one is the non-squeezing property for the flow map of 4NLS in the symplectic phase space $L^2(\mathbb{T})$. To prove the former we used the ideas introduced in [Takaoka and Tsutsumi 2004] and [Nakanish et al 2010], and to prove the latter we used the ideas in [Colliander et al 2005].