Periodic Cubic Hyperbolic Schrödinger equation on \T²

2012/05/23 by Yuzhao Wang, Wang, Yuzhao · 7 citations
Mathematics · #35Q55 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Analysis and Transform Methods #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.1205.5205

openalex publication_date 2012/05/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the cubic Hyperbolic Schrödinger equation \eqrefeq:nls on torus \T². We prove that sharp L⁴ Strichartz estimate, which implies that \eqrefeq:nls is analytic locally well-posed in in H^s(\T²) with s>1/2, meanwhile, the ill-posedness in H^s(\T²) 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.

Cited by

• Sharp L⁴ Strichartz estimate for Hyperbolic Schrödinger equation on ℝ× \mathbbT
• Well-posedness for the periodic Hyperbolic nonlinear Schrödinger equations
• On sharp Strichartz estimate for hyperbolic Schrödinger equation on \mathbbT³
• Sharp unconditional well-posedness of the 2-d periodic cubic hyperbolic nonlinear Schrödinger equation
• On restricted-type Strichartz estimates and the applications
• Essential self-adjointness of real principal type operators
• Strichartz estimates for non elliptic Schrödinger equations on compact manifolds

Related

• Sharp L⁴ Strichartz estimate for Hyperbolic Schrödinger equation on ℝ× \mathbbT
• Well-posedness for the periodic Hyperbolic nonlinear Schrödinger equations
• On sharp Strichartz estimate for hyperbolic Schrödinger equation on \mathbbT³
• Sharp local well-posedness of the two-dimensional periodic nonlinear Schrödinger equation with a quadratic nonlinearity |u|²
• Hyperbolic nonlinear Schrödinger equations on ℝ× \mathbbT