Unconditional well-posendness for the fourth order nonlinear Schrodinger type equations on the torus

TL;DR

This paper establishes the optimal regularity threshold for the unconditional well-posedness of fourth order nonlinear Schrödinger equations on the torus, including non-integrable cases, using normal form reduction and cancellation techniques.

Contribution

It proves the well-posedness for s ≥ 1, extending results to non-integrable cases and identifying the sharp regularity threshold.

Findings

Well-posedness established for s ≥ 1

Includes non-integrable cases

Optimal regularity threshold identified

Abstract

We prove the unconditional well-posedness for the fourth order nonlinear Schrodinger type equations in H^s(\mathbb{T}) when s \geq 1, which includes the non-integrable case. This regularity threshold is optimal because the nonlinear terms cannot be defined in the space-time distribution framework for s<1. The main idea is to employ the normal form reduction and a kind of cancellation property to deal with derivative losses.