An exactly solvable $\mathcal{PT}$-symmetric dimer from a Hamiltonian system of nonlinear oscillators with gain and loss

TL;DR

This paper demonstrates that a pair of nonlinear oscillators with gain and loss can form an exactly solvable Hamiltonian system, leading to a $ ext{PT}$-symmetric nonlinear Schrödinger dimer with softened symmetry-breaking transition.

Contribution

It introduces a Hamiltonian model of coupled nonlinear oscillators with gain and loss, exactly solvable as a $ ext{PT}$-symmetric nonlinear Schrödinger dimer, and analyzes its symmetry-breaking behavior.

Findings

The system is exactly solvable in elementary functions.

Nonlinearity softens the $ ext{PT}$-symmetry breaking transition.

Stable states persist for arbitrarily large gain-loss coefficients.

Abstract

We show that a pair of coupled nonlinear oscillators, of which one oscillator has positive and the other one negative damping of equal rate, can form a Hamiltonian system. Small-amplitude oscillations in this system are governed by a $\mathcal{PT}$-symmetric nonlinear Schr\"odinger dimer with linear and cubic coupling. The dimer also represents a Hamiltonian system and is found to be exactly solvable in elementary functions. We show that the nonlinearity softens the $\mathcal{PT}$-symmetry breaking transition in the nonlinearly-coupled dimer: stable periodic and quasiperiodic states with large enough amplitudes persist for an arbitrarily large value of the gain-loss coefficient.