The defocusing nonlinear Schr\"odinger equation with $t$-periodic data: New exact solutions

TL;DR

This paper characterizes and constructs new exact solutions for the defocusing nonlinear Schrödinger equation on the half-line with large-time periodic boundary data, expanding understanding of its solution space.

Contribution

It provides a theorem that characterizes possible large-time periodic boundary values and offers a constructive method to find explicit solutions.

Findings

Identifies conditions for large-time periodic boundary data

Provides explicit solutions matching these boundary conditions

Expands the set of known exact solutions for the defocusing NLS

Abstract

We consider solutions of the defocusing nonlinear Schr\"odinger (NLS) equation on the half-line whose Dirichlet and Neumann boundary values become periodic for sufficiently large $t$. We prove a theorem which, modulo certain assumptions, characterizes the pairs of periodic functions which can arise as Dirichlet and Neumann values for large $t$ in this way. The theorem also provides a constructive way of determining explicit solutions with the given periodic boundary values. Hence our approach leads to a class of new exact solutions of the defocusing NLS equation on the half-line.