  Coupled Boussinesq equations describe long weakly-nonlinear longitudinal
strain waves in a bi-layer with a soft bonding between the layers (e.g. a soft
adhesive). From the mathematical viewpoint, a particularly difficult case
appears when the linear long-wave speeds in the layers are significantly
different (high-contrast case). The traditional derivation of the
uni-directional models leads to four uncoupled Ostrovsky equations, for the
right- and left-propagating waves in each layer. However, the models impose a
``zero-mass constraint'' i.e. the initial conditions should necessarily have
zero mean, restricting the applicability of that description. Here, we bypass
the contradiction in this high-contrast case by constructing the solution for
the deviation from the evolving mean value, using asymptotic multiple-scale
expansions involving two pairs of fast characteristic variables and two
slow-time variables. By construction, the Ostrovsky equations emerging within
the scope of this derivation are solved for initial conditions with zero mean
while initial conditions for the original system may have non-zero mean values.
Asymptotic validity of the solution is carefully examined numerically. We apply
the models to the description of counter-propagating waves generated by
solitary wave initial conditions, or co-propagating waves generated by cnoidal
wave initial conditions, as well as the resulting wave interactions, and
contrast with the behaviour of the waves in bi-layers when the linear long-wave
speeds in the layers are close (low-contrast case). One local (classical) and
two non-local (generalised) conservation laws of the coupled Boussinesq
equations for strains are derived, and these are used to control the accuracy
of the numerical simulations.
