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 ϵ, and constant linear gain and loss, Y, in each half-box. The system may be readily 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. Their instability mainly leads to blowup, except for the case of ϵ =0, when an unstable symmetric mode transforms into a weakly oscillating breather, and an unstable antisymmetric mode relaxes into a stable symmetric one. At ϵ > 0, the stability area is much larger for the PT -antisymmetric state than for its symmetric counterpart. The stability areas shrink 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 y , and then disappears at a critical value of y.