Nonlinear Schr\"odinger equation on the half-line without a conserved number of solitons

TL;DR

This paper investigates soliton absorption and emission at the boundary of the nonlinear Schrödinger equation on the half-line, revealing non-conservation of charges and proposing a classical boundary algebra framework.

Contribution

It introduces a classical boundary algebra for the nonlinear Schrödinger equation, accounting for soliton boundary interactions and non-conservation of charges.

Findings

Boundary absorption/emission of solitons demonstrated.

Infinite conserved quantities restored with boundary considerations.

Classical boundary algebra derived for the first time.

Abstract

We explore the phenomena of absorption/emission of solitons by an integrable boundary for the nonlinear Schr\"odinger equation on the half-line. This is based on the investigation of time-dependent reflection matrices which satisfy the boundary zero curvature equation. In particular, this leads to absorption/emission processes at the boundary that can take place for solitons and higher-order solitons. As a consequence, the usual charges on the half-line are no longer conserved but we show explicitly how to restore an infinite set of conserved quantities by taking the boundary into account. The Hamiltonian description and Poisson structure of the model are presented, which allows us to derive for the first time a classical version of the boundary algebra used originally in the context of the quantum nonlinear Schr\"odinger equation.