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A. Chernyavsky and D.E. Pelinovsky

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Long-time stability of breathers in Hamiltonian PT-symmetric lattices

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Journal of Physics A: Mathematical Theoretical 49, 475201 (20pp) (2016)

**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
(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 nite time interval. The nonlinear stability
analysis becomes possible for the PT-symmetric lattice only because of the existence of a
Hamiltonian structure.

**Keywords**:

PT-symmetry, discrete nonlinear Schrodinger equation, Hamiltonian structure,
discrete breathers, long-time stability.