  Coupled asymmetric double well ($a\phi^2-b\phi^3+c\phi^4$) one-dimensional
potentials arise in the context of first order phase transitions both in
condensed matter physics and field theory. Here we provide an exhaustive set of
exact periodic solutions of such a coupled asymmetric model in terms of
elliptic functions (domain wall arrays) and obtain single domain wall solutions
in specific limits. We also calculate the energy and interaction between
solitons for various solutions. Both topological (kink-like at $T=T_c$) and
nontopological (pulse-like for $T\ne T_c$) domain wall solutions are obtained.
We relate some of these solutions to domain walls in hydrogen bonded materials
and also in the field theory context. As a byproduct, we also obtain a new one
parameter family of kink solutions of the uncoupled asymmetric double well
model.
