In this lecture we outline 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 a nonlinear equation with a non-smooth term which cannot be linearized even in the limit of very small amplitudes. Finally, we briefly describe the self-similar and shock-wave solutions of that equation.

PACS numbers: 03.50.Kk, 11.10.Lm