  The recently developed formalism of nonlinear fluctuating hydrodynamics
(NLFH) has been instrumental in unraveling many new dynamical universality
classes in coupled driven systems with multiple conserved quantities. In
principle, this formalism requires knowledge of the exact expression of locally
conserved current in terms of local density of the conserved components.
However, for most nonequilibrium systems an exact expression is not available
and it is important to know what happens to the predictions of NLFH in these
cases. We address this question for the first time here in a system with
coupled time evolution of sliding particles on a fluctuating energy landscape.
In the disordered phase this system shows short-ranged correlations, this
system shows short-ranged correlations, the exact form of which is not known,
and so the exact expression for current cannot be obtained. We use approximate
expressions based on mean-field theory and corrections to it, to test the
prediction of NLFH using numerical simulations. In this process we also
discover important finite size effects and show how they affect the predictions
of NLFH. We find that our system is rich enough to show a large variety of
universality classes. From our analytics and simulations we have been able to
find parameter values which lead to diffusive, Kardar-Parisi-Zhang (KPZ), $5/3
$ L\'evy and modified KPZ universality classes. Interestingly, the scaling
function in the modified KPZ case turns out to be close to the Pr\"ahofer-Spohn
function which is known to describe usual KPZ scaling. Our analytics also
predict the golden mean and the $3/2$ L\'evy universality classes within our
model but our simulations could not verify this, perhaps due to strong finite
size effects.
