PT-symmetric and antisymmetric nonlinear states in a split potential box

We introduce a one-dimensional PT-symmetric system, which includes the cubic self-focusing, a double-well potential in the form of an infinitely deep potential box split in the middle by a delta-functional barrier of an effective height epsilon, and constant linear gain and loss, gamma, in each half-box. The setting may be realized in microwave photonics. Using numerical methods, we construct PT-symmetric and antisymmetric modes, which represent, respectively, the system's ground state and first excited state, and identify their stability. When they are unstable, the instability leads to blowup, except for the case of epsilon = 0, when an unstable symmetric mode transforms into a weakly oscillating breather, and an unstable antisymmetric mode relaxes into a stable symmetric one. At epsilon > 0, the stability area is much larger for the PT-antisymmetric state than for its symmetric counterpart. The stability areas shrink with with increase of the total power, P. In the linear limit, which corresponds to P --> 0, the stability boundary is found in an analytical form. The stability area of the antisymmetric state originally expands with the growth of gamma, and then disappears at a critical value of gamma.