I.V. Barashenkov, D.E. Pelinovsky, and Ph. Dubard

Dimer with gain and loss: Integrability and PT-symmetry restoration

Journal of Physics A: Math. Theor. 48, 325201 (28 pages) (2015)

A PT-symmetric dimer is a two-site discrete nonlinear Schrodinger equation with one site losing and the other one gaining energy at the same rate. We construct two four-parameter families of cubic PT-symmetric dimers as gain-loss extensions of their conservative, Hamiltonian, counterparts. Our main result is that, barring a single exceptional case, all these damped-driven nonlinear Schrodinger equations define completely integrable Hamiltonian systems. Furthermore, we identify dimers that exhibit the nonlinearity-induced PT-symmetry restoration. When a dimer of this type is in its symmetry-broken phase, the exponential growth of small initial conditions is saturated by the nonlinear coupling which diverts increasingly large amounts of energy from the gaining to the losing site. As a result, the exponential growth is arrested and all trajectories remain trapped in a finite part of the phase space regardless of the value of the gain-loss coefficient.

PT-symmetry, nonlinear Schrodinger dimer, Hamiltonian structure, integrability, trajectory confinement, spontaneous symmetry restoration.