  We investigate self-similar solutions of evolution equation of a
(1+1)-dimensional field model with the V-shaped potential $U(\phi) = | \phi |,$
where $\phi$ is a real scalar field. The equation contains a nonlinear term of
the form $sign(\phi)$, and it possesses a scaling symmetry. It turns out that
there are several families of the self-similar solutions with qualitatively
different behaviour. We also discuss a rather interesting example of evolution
with non self-similar initial data - the corresponding solution contains a
self-similar component.
