Periodic Cubic Hyperbolic Schrödinger equation on $\T^2$

Abstract

We consider the cubic Hyperbolic Schrödinger equation \eqref{eq:nls} on torus $\T^2$. We prove that sharp $L^4$ Strichartz estimate, which implies that \eqref{eq:nls} is analytic locally well-posed in in $H^s(\T^2)$ with $s>1/2$, meanwhile, the ill-posedness in $H^s(\T^2)$ for $s<1/2$ is also obtained. The main difficulty comes from estimating the number of representations of an integer as a difference of squares.

The reference [6] is added according to Professor Tzvetkov, we also noticed that [8] generalized our main estimates by a different approach recently

References in corpus (2)

• Remark on the periodic mass critical nonlinear Schrödinger equation
• Periodic nonlinear Schrödinger equation in critical $H^s(\T^n)$ spaces