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

Abstract

We consider the Hamiltonian version of a PT -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 (timeperiodic 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 PT -symmetric lattice only because of the existence of a Hamiltonian structure.

Breathers in Hamiltonian PT -Symmetric Chains of Coupled Pendula under a Resonant Periodic Force

“…Every breather along the branch is spectrally stable and is free of resonance between isolated eigenvalues and continuous spectrum. In the follow-up work [12], we will prove long-time orbital stability of breathers along this branch.…”

“…Regarding branch (a), nonlinear stability analysis is not available by using the energy method. Our follow-up work [12] develops a new method of analysis to prove the long-time stability of breathers for branch (a).…”

Krein signature for instability of PT-symmetric states

“…Nixon and Yang [27] considered the linear Schrödinger equation with a complex-valued PT -symmetric potential and introduced the indefinite PT -inner product with the induced PT -Krein signature, in the exact correspondence with the Hamiltonian-Krein signature. In our previous works [11,12], we considered a Hamiltonian version of the PT -symmetric system of coupled oscillators and introduced Krein signature of eigenvalues by using the corresponding Hamiltonian. In the recent works [2,3,35], a coupled non-Hamiltonian PT -symmetric system was considered and the linearized system was shown to be block-diagonalizable to the form where Krein signature of eigenvalues can be introduced.…”

Snakes and ghosts in a parity-time-symmetric chain of dimers

“…Nonlinear stability may be obtained numerically by evolving a perturbed solution in Eqs. (1) for a long while, which analytically is still an open problem due to the absence of a Hamiltonian structure of the system (see, e.g., [44,45] for nonlinear stability analysis of a similar system but with crossdispersion and different nonlinearity that becomes possible because it has a Hamiltonian form via a cross-gradient symplectic structure).…”