  We study the coupled dynamics of the displacement fields in a one dimensional
coupled-field model for drifting crystals, first proposed by R.Lahiri and
S.Ramaswamy [{\em Phys. Rev. Lett.} {\bf 79}, 1150 (1997)]. We present some
exact results for the steady state and the current in the lattice version of
the model, for a special subspace in the parameter space, within the region
where the model displays kinematic waves. We use these to construct the
effective continuum equations corresponding to the lattice model. These
equations decouple at the linear level in terms of the eigenmodes. We examine
the long-time, large-distance properties of the correlation functions of the
eigenmodes by using symmetry arguments, Monte Carlo simulations and
self-consistent mode coupling methods. For most parameter values, the scaling
exponents of the Kardar-Parisi-Zhang equation are obtained. However, for
certain symmetry-determined values of the coupling constants the two
eigenmodes, although nonlinearly coupled, are characterized by two distinct
dynamic exponents. We discuss the possible application of the dynamic
renormalization group in this context.
