Hamiltonian formulation of the standard $\mathcal{PT}$-symmetric nonlinear Schr\"odinger dimer

TL;DR

This paper demonstrates that the standard $ ext{PT}$-symmetric nonlinear Schrödinger dimer, despite involving gain and loss, can be formulated as a Hamiltonian system and provides a corresponding Lagrangian formulation.

Contribution

It introduces a Hamiltonian and Lagrangian formulation for the standard $ ext{PT}$-symmetric dimer, revealing its underlying conservative structure.

Findings

The $ ext{PT}$-symmetric dimer is a Hamiltonian system.

A Lagrangian formulation for the dimer is constructed.

The system maintains a Hamiltonian structure despite gain and loss.

Abstract

The standard $\mathcal{PT}$-symmetric dimer is a linearly-coupled two-site discrete nonlinear Schr\"odinger equation with one site losing and the other one gaining energy at the same rate. We show that despite gain and loss, the standard $\mathcal{PT}$-dimer is a Hamiltonian system. We also produce a Lagrangian formulation for the dimer.