  We consider superstring theories on pp-wave backgrounds which result in an
integrable ${\cal N}=(2,2)$ supersymmetric Landau-Ginzburg theory on the
worldsheet. We obtain exact eigenvalues of the light-cone gauge superstring
hamiltonian in the massive and interacting world-sheet theory with
superpotential $Z^3-Z$. We find the modes of the supergravity part of the
string spectrum, and their space-time interpretation. Because the system is
effectively at strong coupling on the worldsheet, these modes are not in
one-to-one correspondence with the usual type IIB supergravity modes in the
$p_{-} \to 0$ limit. However, the above correspondence holds in the $\alpha'\to
0$ limit.
