  The paper combines two topics belonging to the general theme of the
spontaneous symmetry breaking (SSB) in systems including two basic competing
ingredients: the self-focusing cubic nonlinearity and a double-well-potential
(DWP) structure. Such systems find diverse physical realizations, chiefly in
optical waveguides, made of a nonlinear material and featuring a transverse DWP
structure, and in models of atomic BEC with attractive inter-atomic
interactions, loaded into a pair of symmetric potential wells coupled by
tunneling across the separating barrier. With the increase of the nonlinearity
strength, the SSB occurs at a critical value of the strength. The first part of
the paper offers a brief overview of the topic. The second part presents a
model which is designed as the simplest one capable to produce the SSB
phenomenology in the one-dimensional geometry. The model is based on the DWP
built as an infinitely deep potential box, which is split into two wells by a
delta-functional barrier at the central point. Approximate analytical
predictions for the SSB are produced for two limit cases: strong (deep) or weak
(shallow) splitting of the potential box by the central barrier. Critical
values of the strength of the nonlinearity at the SSB point, represented by the
norm of the stationary wave field, are found in both cases (the critical
strength is small in the former case, and large in the latter one). For the
intermediate case, a less accurate variational approximation (VA) is developed.
