  We introduce domain-wall (DW) states in the bimodal discrete nonlinear
Schr{\"{o}}dinger equation, in which the modes are coupled by cross phase
modulation (XPM). By means of continuation from various initial patterns taken
in the anti-continuum (AC) limit, we find a number of different solutions of
the DW type, for which different stability scenarios are identified. In the
case of strong XPM coupling, DW configurations contain a single mode at each
end of the chain. The most fundamental solution of this type is found to be
always stable. Another solution, which is generated by a different AC pattern,
demonstrates behavior which is unusual for nonlinear dynamical lattices: it is
unstable for small values of the coupling constant $C$ (which measures the
ratio of the nonlinearity and coupling lengths), and becomes stable at larger
$C$. Stable bound states of DWs are also found. DW configurations generated by
more sophisticated AC patterns are identified as well, but they are either
completely unstable, or are stable only at small values of $C$. In the case of
weak XPM, a natural DW solution is the one which contains a combination of both
polarizations, with the phase difference between them 0 and $\pi $ at the
opposite ends of the lattice. This solution is unstable at all values of $C$,
but the instability is very weak for large $C$, indicating stabilization as the
continuum limit is approached. The stability of DWs is also verified by direct
simulations, and the evolution of unstable DWs is simulated too; in particular,
it is found that, in the weak-XPM system, the instability may give rise to a
moving DW.
