  In this lecture we outline the main results of our investigations of certain
field-theoretic systems which have V-shaped field potential. After presenting
physical examples of such systems, we show that in static problems the exact
ground state value of the field is achieved on a finite distance - there are no
exponential tails. This applies in particular to soliton-like object called the
topological compacton. Next, we discuss scaling invariance which appears when
the fields are restricted to small amplitude perturbations of the ground state.
Evolution of such perturbations is governed by nonlinear equation with a
non-smooth term which can not be linearized even in the limit of very small
amplitudes. Finally, we briefly describe self-similar and shock wave solutions
of that equation.
