On the quasilinear Schr\"odinger equations on tori

TL;DR

This paper establishes well-posedness for quasilinear Hamiltonian Schrödinger equations on tori in high regularity Sobolev spaces, improving previous results by leveraging advanced paradifferential calculus techniques.

Contribution

It improves the regularity threshold for well-posedness of quasilinear Schrödinger equations on tori using sharp paradifferential calculus methods.

Findings

Well-posedness established for s > d/2 + 3.

Enhanced regularity results compared to previous work.

Application of advanced paradifferential calculus on tori.

Abstract

We improve the result by Feola and Iandoli [J. de Math. Pures et App., 157:243-281, 2022], showing that quasilinear Hamiltonian Schr\"odinger type equations are well posed on $H^s(\mathbb{T}^d)$ if $s>d/2+3$. We exploit the sharp paradifferential calculus on $\mathbb{T}^d$ introduced by Berti, Maspero and Murgante [J. Dynam. and Differential Equations, 33 (3): 1475-1513, 2021].