Hamiltonian formulation of the standard $\mathcal{PT}$-symmetric nonlinear Schrödinger dimer

The standard $\mathcal{PT}$-symmetric dimer is a linearly-coupled two-site discrete nonlinear Schrödinger equation with one site losing and the other one gaining energy at the same rate. We show that despite gain and loss, the standard $\mathcal{PT}$-dimer is a Hamiltonian system. We also produce a Lagrangian formulation for the dimer.

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