Symmetric hyperbolic Schr\"{o}dinger equations on tori

TL;DR

This paper establishes sharp Strichartz estimates for symmetric hyperbolic Schrödinger equations on tori and proves local well-posedness for specific hyperbolic nonlinear Schrödinger equations in two dimensions.

Contribution

It extends Strichartz estimates to general tori using Bourgain's method and demonstrates optimal local well-posedness for septic HNLS and hyperbolic-elliptic Davey-Stewartson systems.

Findings

Sharp Strichartz estimates on general tori, especially rational tori.

Optimal local well-posedness for septic HNLS.

Well-posedness results for hyperbolic-elliptic Davey-Stewartson system.

Abstract

In this paper, we study the symmetric hyperbolic Schr\"{o}dinger equations in the periodic setting. First, we prove full range Strichartz estimates on general tori by adapting Bourgain's major arc method. The result is sharp for rational tori. Second, on two-dimensional rational tori, we establish optimal local well-posedness for two hyperbolic nonlinear Schr\"{o}dinger (HNLS) equations: the septic HNLS and the hyperbolic-elliptic Davey-Stewartson system.