The Effects of Nonlinear Noise on the Fractional Schrödinger Equation

Abstract

The aim of this work is to investigate the influence of nonlinear multiplicative noise on the Cauchy problem of the nonlinear fractional Schrödinger equation in the non-radial case. Local well-posedness follows from estimates related to the stochastic convolution and deterministic non-radial Strichartz estimates. Furthermore, the blow-up criterion is presented. Then, with the help of Itô’s lemma and stopping time arguments, the global solution is constructed almost surely. The main innovation is that the non-radial global solution is given under fractional-order derivatives and a nonlinear noise term.

Keywords

• nonlinear fractional Schrödinger equation
• non-radial Strichartz estimates
• local well-posedness
• Itô’s lemma
• global solution