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)’.