Nonlinear dynamics in PT-symmetric lattices

Abstract

We consider nonlinear dynamics in a finite parity-time-symmetric chain of the discrete nonlinear Schr{ö}dinger (dNLS) type. We work in the range of the gain and loss coefficient when the zero equilibrium state is neutrally stable. We prove that the solutions of the dNLS equation do not blow up in a finite time and the trajectories starting with small initial data remain bounded for all times. Nevertheless, for arbitrary values of the gain and loss parameter, there exist trajectories starting with large initial data that grow exponentially fast for larger times with a rate that is rigorously identified. Numerical computations illustrate these analytical results for dimers and quadrimers.

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Cited by in corpus (25)

• Nonlinear waves in $\mathcal{P}\mathcal{T}$-symmetric systems
• Nonlinear switching and solitons in PT-symmetric photonic systems
• Hamiltonian formulation of the standard $\mathcal{P}\mathcal{T}$-symmetric nonlinear Schrödinger dimer
• PT-Symmetric Dimer of Coupled Nonlinear Oscillators
• Dimer with gain and loss: Integrability and $\mathcal{P}\mathcal{T}$-symmetry restoration
• On the integrability of PT-symmetric dimers
• Jamming anomaly in $\mathcal{P}\mathcal{T}$-symmetric systems
• Nonlinear modes in a generalized $\mathcal{P}\mathcal{T}$-symmetric discrete nonlinear Schrödinger equation
• Revisiting the optical $PT$-symmetric dimer
• Solitons in PT-symmetric ladders of optical waveguides
• Interaction of sine-Gordon kinks and breathers with a $\mathcal{P}\mathcal{T}$-symmetric defect
• Stationary modes and integrals of motion in nonlinear lattices with PT-symmetric linear part
• Stochastic $\mathcal{P}\mathcal{T}$-symmetric coupler
• $\mathcal{P}\mathcal{T}$-symmetric coupler with ${\chi }^{(2)}$ nonlinearity
• Stabilization of the coupled pendula chain under parametric PT-symmetric driving force
• PT-symmetry Management in Oligomer Systems
• Long-time stability of breathers in Hamiltonian $\mathcal{P}\mathcal{T}$-symmetric lattices
• Generalized Dimers and their Stokes-variable Dynamics
• Global existence of solutions to coupled $\mathcal{P}\mathcal{T}$-symmetric nonlinear Schrödinger equations
• Non-Hermitian dynamics of a two-spin system with PT symmetry
• Nonlinear plasmonic amplification via dissipative soliplasmons
• Integrability and trajectory confinement in $\mathcal{P}\mathcal{T}$-symmetric waveguide arrays
• PT-symmetry in quasi-integrable models
• Nonlinear modes and symmetries in linearly-coupled pairs of PT-invariant dimers
• Existence and stability of PT-symmetric vortices in nonlinear two-dimensional square lattices