Self-similarity for V-shaped field potentials - further examples

Three new models with V-shaped field potentials $U$ are considered: a complex scalar field $X$ in 1+1 dimensions with $U(X)=\mathrm{\mid }X\mathrm{\mid }$, a real scalar field $\mathrm{\Phi }$ in 2+1 dimensions with $U(\mathrm{\Phi })=\mathrm{\mid }\mathrm{\Phi }\mathrm{\mid }$, and a real scalar field $\varphi$ in 1+1 dimensions with U{\phi) = \phi \Theta(\phi) where $\mathrm{\Theta }$ 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.