The nonlinear Schr\"odinger equation with $t$-periodic data: I. Exact results

TL;DR

This paper analyzes the nonlinear Schrödinger equation on a half-line with periodic boundary data, deriving conditions for boundary functions and introducing methods to construct solutions using Riemann-Hilbert and perturbative techniques.

Contribution

It provides a verifiable condition for periodic boundary data and introduces two methods for constructing boundary functions in the nonlinear Schrödinger equation context.

Findings

Derived a verifiable condition for periodic boundary functions.

Developed a Riemann-Hilbert problem formulation for the boundary data.

Presented a perturbative approach for constructing boundary functions.

Abstract

We consider the nonlinear Schr\"odinger 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 $g_0^b(t)$ and $g_1^b(t)$ must satisfy. Furthermore, we introduce two different methods, one based on the formulation of a Riemann-Hilbert problem, and one based on a perturbative approach, for constructing $g_1^b(t)$ ($g_0^b(t)$) in terms of $g_0^b(t)$ ($g_1^b(t)$).