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.

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

References in corpus (7)

• Making Sense of Non-Hermitian Hamiltonians
• Unidirectional Nonlinear PT-symmetric Optical Structures
• Mean-field dynamics of a non-Hermitian Bose-Hubbard dimer
• Quantum Classical Correspondence for a non-Hermitian Bose-Hubbard Dimer
• Breathers in PT-symmetric optical couplers
• Symmetry Breaking in Symmetric and Asymmetric Double-Well Potentials
• Stationary states of a PT-symmetric two-mode Bose-Einstein condensate

Cited by in corpus (8)

• 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
• 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
• Classical Hamiltonian Systems with Balanced loss and gain
• Solitary waves in the resonant nonlinear Schrödinger equation: stability and dynamical properties