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

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

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

**Abstract:**

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.

**Keywords**:

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