The nonlinear Schrödinger equation with t -periodic data: I. Exact results

Abstract

We consider the nonlinear Schrödinger equation on the half-line with a given Dirichlet (Neumann) boundary datum which for large t tends to the periodic function g 0 b ( t ) ( g 1 b ( t ) ). Assuming that the unknown Neumann (Dirichlet) boundary value tends for large t to a periodic function g 1 b ( t ) ( g 0 b ( t ) ), we derive an easily verifiable condition that the functions g 1 b ( t ) and g 0 b ( t ) must satisfy. Furthermore, we propose two different methods, one based on the formulation of a Riemann–Hilbert problem and the other based on a perturbative approach, for constructing g 1 b ( t ) ( g 0 b ( t ) ) in terms of g 0 b ( t ) ( g 1 b ( t ) ).