Existence and Uniqueness theory for the fractional Schrödinger equation on the torus

Abstract

We study the Cauchy problem for the $1$-d periodic fractional Schrödinger equation with cubic nonlinearity. In particular we prove local well-posedness in Sobolev spaces, for solutions evolving from rough initial data. In addition we show the existence of global-in-time infinite energy solutions. Our tools include a new Strichartz estimate on the torus along with ideas that Bourgain developed in studying the periodic cubic NLS.

19 pages

References in corpus (3)

• Well-posedness and Ill-posedness for the cubic fractional Schrödinger equations
• Talbot effect for the cubic nonlinear Schrödinger equation on the torus
• Bounds on Sobolev norms for the nonlinear Schrödinger equation on general tori

Cited by in corpus (6)

• Strichartz estimates for the fractional Schrödinger and wave equations on compact manifolds without boundary
• On the transport of Gaussian measures under the one-dimensional fractional nonlinear Schrödinger equations
• Well-posedness and Ill-posedness for the cubic fractional Schrödinger equations
• Global well-posedness and long-time behavior of the fractional NLS
• The cubic nonlinear fractional Schrödinger equation on the half-line
• Smoothing for the Zakharov & Klein-Gordon-Schrödinger Systems on Euclidean Spaces