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

Abstract

We introduce a one-dimensional -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, γ , in each half-box. The system may be readily realized in microwave photonics. Using numerical methods, we construct -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 -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 , the stability boundary is found in an analytical form. The stability area of the antisymmetric state originally expands with the growth of γ , and then disappears at a critical value of γ . This article is part of the theme issue ‘Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 1)’.