Hamiltonian formulation of the standard $\mathcal{P}\mathcal{T}$-symmetric nonlinear Schrödinger dimer

Abstract

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

arXiv comment5 pages; presentation at the 3rd Dynamics Days South America (Viña del Mar, Chile)

References in corpus (26)

• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• Making Sense of Non-Hermitian Hamiltonians
• Quantum Coherent Atomic Tunneling between Two Trapped Bose-Einstein Condensates
• Pseudo-Hermitian Representation of Quantum Mechanics
• Unidirectional Nonlinear PT-symmetric Optical Structures
• Introduction to PT-Symmetric Quantum Theory
• Stability of solitons in PT-symmetric couplers
• The Gross-Pitaevskii Equation for Bose Particles in a Double Well Potential: Two Mode Models and Beyond
• Nonlinear suppression of time-reversals in PT-symmetric optical couplers
• Optical solitons in $\mathcal{P}\mathcal{T}$-symmetric nonlinear couplers with gain and loss
• Mean-field dynamics of a non-Hermitian Bose-Hubbard dimer
• Coupled-mode theory for Bose-Einstein condensates
• Nonlinear modes in finite-dimensional PT-symmetric systems
• Twofold Transition in PT-Symmetric Coupled Oscillators
• Quantum Classical Correspondence for a non-Hermitian Bose-Hubbard Dimer
• PT-Symmetric Oligomers: Analytical Solutions, Linear Stability and Nonlinear Dynamics
• Quantum Lattice Solitons
• Breathers in PT-symmetric optical couplers
• Solitons in a chain of PT-invariant dimers
• Symmetry Breaking in Symmetric and Asymmetric Double-Well Potentials
• Stationary states of a PT-symmetric two-mode Bose-Einstein condensate
• Linear and Nonlinear PT-symmetric Oligomers: A Dynamical Systems Analysis
• Nonlinear dynamics in PT-symmetric lattices
• PT-Symmetric Dimer of Coupled Nonlinear Oscillators
• On the integrability of PT-symmetric dimers
• PT-symmetry Management in Oligomer Systems

Cited by in corpus (22)

• Nonlinear waves in $\mathcal{P}\mathcal{T}$-symmetric systems
• Nonlinear switching and solitons in PT-symmetric photonic systems
• Giant nonlinearity via breaking parity-time symmetry: a route to low-threshold phonon diodes
• Dimer with gain and loss: Integrability and $\mathcal{P}\mathcal{T}$-symmetry restoration
• On integrable wave interactions and Lax pairs on symmetric spaces
• On the N-wave Equations with PT-symmetry
• Jamming anomaly in $\mathcal{P}\mathcal{T}$-symmetric systems
• Revisiting the optical $PT$-symmetric dimer
• Light dynamics in nonlinear trimers and twisted multicore fibers
• Nonlocal reductions of the Ablowitz-Ladik equation
• Nonlinear anti-directional couplers with gain and loss
• On nonlocal reductions of the multi-component nonlinear Schrodinger equation on symmetric spaces
• Bistable soliton switching dynamics in a $\mathcal{P}\mathcal{T}$-symmetric coupler with saturable nonlinearity
• Dynamical control of solitons in a parity-time-symmetric coupler by periodic management
• Non-Hermitian dynamics of a two-spin system with PT symmetry
• Systems that become $\mathcal{P}\mathcal{T}$ symmetric through interaction
• Solitons in a Hamiltonian $PT$-symmetric coupler
• Integrability and trajectory confinement in $\mathcal{P}\mathcal{T}$-symmetric waveguide arrays
• Robust PT symmetry of two-dimensional fundamental and vortex solitons supported by spatially modulated nonlinearity
• Making the PT symmetry unbreakable
• Classical Hamiltonian Systems with Balanced loss and gain
• Solitary waves in the resonant nonlinear Schrödinger equation: stability and dynamical properties