Inhomogeneous initial-boundary value problem for the 2D nonlinear Schrödinger equation

Abstract

We consider the inhomogeneous Dirichlet initial-boundary value problem for the nonlinear multidimensional Schrödinger equation, formulated on an upper right-quarter plane. We study traditionally important problems of the theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial-boundary value problem and the asymptotic behavior of solutions for large time. Also we are interested in the study of the influence of the Dirichlet boundary data on the asymptotic behavior of solutions. Our approach to get well-posedness of nonlinear problems is based on studying a linear theory and then using the fixed point argument. To get a linear theory for the multidimensional model, we proposed a general method based on the Riemann–Hilbert approach and theory Cauchy type integral equations. To get smooth solutions in L∞, we modify a method based on the factorization for the free Schrödinger evolution group.