Type: Article

Multidimensional inhomogeneous mixed initial-boundary value problem for the nonlinear Schrödinger equation

Arciga-Alexandre, Martín P; Kaikina, E. I.;

Abstract:

We consider the inhomogeneous mixed initial-boundary value problem for the nonlinear multidimensional Schrödinger equation, formulated on 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 mixta boundary data on the asymptotic behavior of solutions. Our approach to get well-posedness of nonlinear problems is based on the studying a linear theory and then using the fixed point argument. To get a linear theory for the multidimensional model we propose general method based on Laplace 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. The advantage of our approach is that it can also be applied to non-integrable equations and arbitrary boundary conditions. This approach is new and it is not standard. We believe that the results of this paper could be applicable to study a wide class of dissipative multidimensional nonlinear equations by the use of techniques of nonlinear analysis.