Nonlinear fractional Schrödinger equation on the half-Line
Tóm tắt
This paper study the local-in-time existence of solution for the nonlinear fractional Schrödinger equation with Levy indices
$$1<\alpha <2$$
on the half-line
$$i\partial _t u+(-\Delta ^+)^{\alpha /2}u+\lambda u|u|^{p-1}=0$$
, for measurable initial data and inhomogeneous Dirichlet boundary condition. The mathematical analysis developed in this paper combine two different aspects, the Riemann–Liouville fractional by Colliander and Kenig (Commun Partial Differ Equ 27:2187-2266, 2002) and Fourier restriction method of Sobolev space by Holmer (Differ Integr Equ 18:647-668, 2005) and Holmer (Uniform Estimates for the Zakharov System and the Initial-boundary Value Problem for the Korteweg-de Vries and Nonlinear Schrödinger Equations, 2004).
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