Neumann problem for nonlinear Schrödinger equation with the Riesz fractional derivative operator

We consider the initial–boundary value problem for the fractional Schrödinger equation, posed on positive half-line x>0x>0:{ut+iuxx+i|u|2u+|∂x|12u=0,t≥0,x≥0;u(x,0)=u0(x),x>0,ux(0,t)=h(t),t>0, where |∂x|12 is the fractional derivative operator defined by the Riesz potential|∂x|12=12π∫0∞sign(x−y)|x−y|uy(y)dy. We study the global existence in time and asymptotics of solutions to the initial–boundary value problem.