  Three new models with V-shaped field potentials $U$ are considered: a complex
scalar field $X$ in 1+1 dimensions with $U(X)= |X|$, a real scalar field $\Phi$
in 2+1 dimensions with $U(\Phi) = |\Phi|$, and a real scalar field $\phi$ in
1+1 dimensions with $U{\phi) = \phi \Theta(\phi)$ where $\Theta$ is the step
function. Several explicit, self-similar solutions are found. They describe
interesting dynamical processes, for example, `freezing' a string in a static
configuration.
