An exactly solvable -symmetric dimer from a Hamiltonian system of nonlinear oscillators with gain and loss

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We show that a pair of coupled nonlinear oscillators, of which one oscillator has positive and the other one negative damping of equal rate, can form a Hamiltonian system. Small-amplitude oscillations in this system are governed by a -symmetric nonlinear Schrödinger dimer with linear and cubic coupling. The dimer also represents a Hamiltonian system and is found to be exactly solvable in elementary functions. We show that the nonlinearity softens the -symmetry breaking transition in the nonlinearly-coupled dimer: stable periodic and quasiperiodic states with large enough amplitudes persist for an arbitrarily large value of the gain–loss coefficient.