Strings on pp-waves and massive two dimensional field theories

We find a general class of pp-wave solutions of type IIB string theory such that the light cone gauge worldsheet lagrangian is that of an interacting massive field theory. When the light cone Lagrangian has (2,2) supersymmetry we can find backgrounds that lead to arbitrary superpotentials on the worldsheet. We consider situations with both flat and curved transverse spaces. We describe in some detail the background giving rise to the N=2 sine Gordon theory on the worldsheet. Massive mirror symmetry relates it to the deformed $CP^1$ model (or sausage model) which seems to elude a purely supergravity target space interpretation.