Long-time stability of breathers in Hamiltonian $\mathcal{P}\mathcal{T}$-symmetric lattices

Abstract

We consider the Hamiltonian version of a $\mathcal{P}\mathcal{T}$-symmetric lattice that describes dynamics of coupled pendula under a resonant periodic force. Using the asymptotic limit of a weak coupling between the pendula, we prove the nonlinear long-time stability of breathers (time-periodic solutions localized in the lattice) by using the Lyapunov method. Breathers are saddle points of the extended energy function, which are located between the continuous bands of positive and negative energy. Nevertheless, we construct an approximate Lyapunov function and estimate its evolution on a long but finite time interval. The nonlinear stability analysis becomes possible for the $\mathcal{P}\mathcal{T}$-symmetric lattice only because of the existence of a Hamiltonian structure.

References in corpus (3)

• Making Sense of Non-Hermitian Hamiltonians
• Breathers in PT-symmetric optical couplers
• Global existence of solutions to coupled $\mathcal{P}\mathcal{T}$-symmetric nonlinear Schrödinger equations

Cited by in corpus (6)

• Solitons in PT-symmetric ladders of optical waveguides
• Spinor solitons and their $\mathcal{P}\mathcal{T}$-symmetric offspring
• Snakes and ghosts in a parity-time-symmetric chain of dimers
• Stabilization of the coupled pendula chain under parametric PT-symmetric driving force
• Solitons in a Hamiltonian $PT$-symmetric coupler
• Krein signature for instability of $\mathcal{P}\mathcal{T}$-symmetric states