Hyperbolic nonlinear Schrödinger equations on $\mathbb{R}\times \mathbb{T}$
Engin Başakoğlu, Chenmin Sun, Nikolay Tzvetkov, Yuzhao Wang
TL;DR
This work analyzes the hyperbolic nonlinear Schrödinger equation on the semi‑periodic manifold $\mathbb{R}\times\mathbb{T}$, proving sharp local well‑posedness at the critical regularity $H^{1-\frac{1}{k}}$ for $k\ge2$ (and local well‑posedness for any $s>0$ when $k=1$). For small initial data, it establishes global well‑posedness and scattering for higher odd nonlinearities (excluding the cubic case) in these critical spaces, by leveraging sharp Strichartz estimates up to the endpoint on $\mathbb{R}\times\mathbb{T}$ and a robust function space framework built from $U^p_{\Box}$, $V^p_{\Box}$, $X^s$, and $Y^s$. A central technical contribution is the endpoint Strichartz theory achieved via an $\varepsilon$‑removal argument adapted to the semi‑periodic setting, combined with global in time estimates through a Barron–Višan–Pausader style analysis. The results extend small data theory known on $\mathbb{R}^2$ to the product domain, providing sharp, scalable tools for the long‑time dynamics and laying groundwork for further improvements in cubic cases via modified scattering techniques. Overall, the paper advances critical‑space well‑posedness and scattering for HNLS on $\mathbb{R}\times\mathbb{T}$ and enriches the Strichartz toolbox for hyperbolic dispersive equations on semi‑periodic manifolds.
Abstract
In this paper, we consider the hyperbolic nonlinear Schrödinger equations (HNLS) on $\mathbb{R}\times\mathbb{T}$. We obtain the sharp local well-posedness up to the critical regularity for cubic nonlinearity and in critical spaces for higher odd nonlinearities. Moreover, when the initial data is small, we prove the global existence and scattering for the solutions to HNLS with higher nonlinearities (except the cubic one) in critical Sobolev spaces. The main ingredient of the proof is the sharp up to the endpoint local/global-in-time Strichartz estimates.