Well-posedness for the periodic Hyperbolic nonlinear Schrödinger equations

TL;DR

The paper proves local well-posedness for periodic hyperbolic nonlinear Schrödinger equations in critical Sobolev spaces by deriving scale-invariant Strichartz estimates on the torus for the hyperbolic Laplacian. It adapts the Killip–Visan framework, employing the $U^p/V^p$-based $X^s$/$Y^s$ spaces, a dispersive analysis of the hyperbolic kernel, and a hyperbolic Galilean boost to obtain cube-localized estimates. The authors establish subcritical and critical nonlinear bounds, enabling a contraction mapping in $X^{s_c(d,m)}$ and proving local well-posedness for a wide range of dimensions and nonlinearities, with almost complete coverage of the hyperbolic regime except for the case $(d,m)=(2,3)$. This work extends the torus Strichartz theory to hyperbolic settings and provides a robust framework for periodic HNLS with derivative-free, scale-invariant control. The results have potential implications for nonlinear dispersive dynamics on compact manifolds and related stability analyses.

Abstract

We establish local well-posedness for the hyperbolic nonlinear Schrodinger equation (HNLS) in the critical spaces. Following the approach of Killip and Visan, we derive scale-invariant Strichartz estimates for HNLS on both rational and irrational tori, thereby removing the epsilon-loss of derivative present in the hyperbolic Strichartz estimates of Bourgain and Demeter.

Theorems & Definitions (50)

• Theorem 1.1: Theorem 2.4 in BD15 and Corollary 1.3 in BD17
• Remark 1.2
• Theorem 1.3: Theorem 2 in KV14
• Theorem 1.4
• Remark 1.5
• Remark 1.6
• Theorem 1.7: elliptic case
• Theorem 1.8: hyperbolic case: subcritical
• Remark 1.9
• Remark 1.10