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 \varphi in 1+1 dimensions with U(\varphi ) = \varphi \Theta (\varphi ) 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.

PACS numbers: 05.45.--a, 03.50.Kk, 11.10.Lm