# Construction of solutions of the defocusing nonlinear Schr\"odinger equation with asymptotically time-periodic boundary values

## TL;DR

This paper constructs solutions to the defocusing nonlinear Schrödinger equation with boundary conditions that become periodic over time, using Riemann-Hilbert problems and steepest descent methods, and analyzes their long-term behavior.

## Contribution

It introduces a method to explicitly construct solutions with asymptotically periodic boundary data and determines their detailed long-time asymptotics.

## Key findings

- Solutions exhibit a leading exponential plane wave behavior.
- Subleading terms in long-time asymptotics are explicitly computed.
- Method applies to quarter-plane problems with asymptotically periodic boundaries.

## Abstract

We study the defocusing nonlinear Schr\"odinger equation in the quarter plane with asymptotically periodic boundary values. By studying an associated Riemann-Hilbert problem and employing nonlinear steepest descent arguments, we construct solutions in a sector close to the boundary whose leading behaviour is described by a single exponential plane wave. Furthermore, we compute the subleading terms in the long time asymptotics of the constructed solutions.