Periodic Cubic Hyperbolic Schr\"odinger equation on $\T^2$

Yuzhao Wang

Periodic Cubic Hyperbolic Schr\"odinger equation on $\T^2$

Analysis of PDEs 2013-04-23 v3

Authors: Yuzhao Wang

Abstract

We consider the cubic Hyperbolic Schr\"odinger 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.

Keywords

nonlinear schrödinger equation schrödinger equation

Cite

@article{arxiv.1205.5205,
  title  = {Periodic Cubic Hyperbolic Schr\"odinger equation on $\T^2$},
  author = {Yuzhao Wang},
  journal= {arXiv preprint arXiv:1205.5205},
  year   = {2013}
}

Comments

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

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