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The dilation method is an important and useful way in experimentally simulating non-Hermitian, especially $cal PT$-symmetric systems. However, the time dependent dilation problem cannot be explicitly solved in general. In this paper, we consider a special two dimensional time dependent $cal PT$-symmetric system, which is initially set in the unbroken $cal PT$-symmetric phase and later goes across the exceptional point and enters the broken $cal PT$-symmetric phase. For this system, the dilation Hamiltonian and the evolution of $cal PT$-symmetric system are analytically worked out.
Three ways of constructing a non-Hermitian matrix with possible all real eigenvalues are discussed. They are PT symmetry, pseudo-Hermiticity, and generalized PT symmetry. Parameter counting is provided for each class. All three classes of matrices ha
This note examines Gross-Pitaevskii equations with PT-symmetric potentials of the Wadati type: $V=-W^2+iW_x$. We formulate a recipe for the construction of Wadati potentials supporting exact localised solutions. The general procedure is exemplified b
Non-hermitian, $mathcal{PT}$-symmetric Hamiltonians, experimentally realized in optical systems, accurately model the properties of open, bosonic systems with balanced, spatially separated gain and loss. We present a family of exactly solvable, two-d
We introduce the simplest one-dimensional nonlinear model with the parity-time (PT) symmetry, which makes it possible to find exact analytical solutions for localized modes (solitons). The PT-symmetric element is represented by a point-like (delta-fu
We consider the linear and nonlinear Schrodinger equation for a Bose-Einstein condensate in a harmonic trap with $cal {PT}$-symmetric double-delta function loss and gain terms. We verify that the conditions for the applicability of a recent propositi