The initial-boundary value problem for the Schrödinger equation with the nonlinear Neumann boundary condition on the half-plane

We consider the initial-boundary value problem of the nonlinear Schrödinger equation on the half plane with a nonlinear Neumann boundary condition. After establishing the boundary Strichartz estimate in \(L^2({\mathbb {R}}^2_+)\) $L^2({\mathbb{R}}_{+}^2)$ and \(H^s({\mathbb {R}}^2_+)\) $H^s({\mathbb{R}}_{+}^2)$ , we consider the time local well-posedness of the problem in \(L^2({\mathbb {R}}^2_+)\) $L^2({\mathbb{R}}_{+}^2)$ and \(H^s({\mathbb {R}}^2_+)\) $H^s({\mathbb{R}}_{+}^2)$ .