Nonlinear Schr\"odinger equation on the half-line with nonlinear boundary condition

In this paper, we study the initial boundary value problem for nonlinear Schr\"odinger equations on the half-line with nonlinear boundary conditions of type $u_x(0,t)+\lambda|u(0,t)|^ru(0,t)=0,$ $\lambda\in\mathbb{R}-\{0\}$, $r> 0$. We discuss the local well-posedness when the initial data $u_0=u(x,0)$ belongs to an $L^2$-based inhomogeneous Sobolev space $H^s(\mathbb{R}_+)$ with $s\in \left(\frac{1}{2},\frac{7}{2}\right)-\{\frac{3}{2}\}$. We deal with the nonlinear boundary condition by first studying the linear Schr\"odinger equation with a time-dependent inhomogeneous Neumann boundary condition $u_x(0,t)=h(t)$ where $h\in H^{\frac{2s-1}{4}}(0,T)$. This latter problem is studied by adapting the method of Bona-Sun-Zhang \cite{BonaSunZhang2015} to the case of inhomogeneous Neumann boundary conditions.