Asymptotics for inhomogeneous Dirichlet initial-boundary value problem for the nonlinear Schrödinger equation

Abstract

We consider the inhomogeneous Dirichlet initial-boundary value problem for the nonlinear Schrödinger equation, formulated on a half-line. 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.

Modified scattering for the mixed initial-boundary problem for the nonlinear Klein–Gordon equation

“…In these papers, the existence of modified scattering for the Dirichlet and Neumann problems for the Klein-Gordon equation with cubic nonlinearities on half-line were considered. We also mention the papers [27][28][29] for related problems in the case of the nonlinear Schrödinger equation and [45,46] for the case of the Thirring model.…”

Nonlinear Schrödinger equation on the half-line with nonlinear boundary condition

“…Although nonlinear Schrödinger equations with inhomogeneous boundary conditions have been studied to some extent, most of these papers were devoted to inhomogeneous Dirichlet boundary conditions; see [10], [4], [5], [8], [6], [26], [9], [14], [21], [16], [3], [22], [24]. There are relatively less results on inhomogeneous Neumann boundary conditions; see [5], [7], [15], [23], [24].…”

Inhomogeneous Dirichlet boundary value problem for nonlinear Schrödinger equations in the upper half-space

“…Another method for analyzing one dimensional initial-boundary value problems, based on the Riemann-Hilbert approach, was introduced in [20]. By this method, in [21], it was shown that long range scattering occurs in CNLS equation. The advantage of Riemann-Hilbert approach is that it can be applied to non-integrable equations with general inhomogeneous boundary data, but some technical problems have to be overcame (see [22,[24][25][26]).…”