Local wellposedness for the critical nonlinear Schr\"odinger equation on $\mathbb{T}^3$

TL;DR

This paper establishes local well-posedness for the critical nonlinear Schrödinger equation on a 3-torus with irrational geometry, extending previous results to a broader class of nonlinearities and initial data regularity.

Contribution

It proves local well-posedness for NLS on irrational 3-tori at critical regularity for all p ≥ 2, generalizing prior work limited to even integer p.

Findings

Well-posedness established at critical regularity s_c for all p ≥ 2.

Extension of local Cauchy theory to irrational tori.

Broader class of nonlinearities covered compared to previous studies.

Abstract

For $p\geq 2$, we prove local wellposedness for the nonlinear Schr\"odinger equation $(i\partial_t + \Delta)u = \pm|u|^pu$ on $\mathbb{T}^3$ with initial data in $H^{s_c}(\mathbb{T}^3)$, where $\mathbb{T}^3$ is a rectangular irrational $3$-torus and $s_c = \frac{3}{2} - \frac{2}{p}$ is the scaling-critical regularity. This extends work of earlier authors on the local Cauchy theory for NLS on $\mathbb{T}^3$ with power nonlinearities where $p$ is an even integer.