  We introduce a model of media with the cubic attractive nonlinearity
concentrated along a single or double stripe in the two-dimensional (2D) plane.
The model can be realized in terms of nonlinear optics (in the spatial and
temporal domains alike) and BEC. In recent works, it was concluded that search
for stable 2D solitons in models with a spatially localized self-attractive
nonlinearity is a challenging problem. We make use of the variational
approximation (VA) and numerical methods to investigate conditions for the
existence and stability of solitons in the present setting. The result
crucially depends on the transverse shape of the stripe: while the rectangular
profile supports stable 2D solitons, its smooth Gaussian-shaped counterpart
makes all the solitons unstable. The double stripe with the rectangular profile
admit stable solitons of three distinct types: symmetric and asymmetric ones
with a single peak, and double-peak symmetric solitons. The shape and stability
of single-peak solitons of either type are accurately predicted by the VA.
Collisions between stable solitons are briefly considered too, by means of
direct simulations. Depending on the relative velocity we observe excitation,
decay or catastrophic self focusing.
