Robin initial-boundary value problem for nonlinear Schrodinger equation with potential

Abstract

We study the initial-boundary value problem for the Schrödinger equation with potential, posed on positive half-line $x>0:$

$$\begin{aligned} \partial _{t}u+ia_{1}\partial _{xx}u+ia_{2}|u|^{2}u+\alpha _{3}\mathcal {P}^{ \frac{1}{2}}u=0, \end{aligned}$$

where $a_{1}\in \mathbb {R},a_{2}\in \mathbb {C}$, $a_{3}>0,$ $\mathcal {P}^{ \frac{1}{2}}$ is the fractional derivative operator defined by the modified Risz potential

$$\begin{aligned} \mathcal {P}^{\frac{1}{2}}u=\frac{1}{2\pi }\int \limits _{0}^{\infty }\mathrm{dy}\frac{ \mathrm{sign}\left( x-y\right) }{|x-y|^{\frac{1}{2}}}\partial _{y}u\left( y,t\right) . \end{aligned}$$

We are interested in the initial-boundary value problem with small initial data $u(x,0)=u_{0}(x)$ and Robin boundary data $u(0,t)+\beta u_{x}(0,t)=h(t),\beta >0,$ given in a suitable weighted Sobolev spaces. We show that problem admits global solutions whose long-time behavior essentially depends on the boundary data.