  In this paper we are consider radiating solitary wave solutions of coupled
regularised Boussinesq equations. This type of solution consists of a leading
solitary wave with a small-amplitude co-propagating oscillatory tail, and
emerges from a pure solitary wave solution of a symmetric reduction of the full
system. We construct an asymptotic solution, where the leading order
approximation in both components is obtained as a particular solution of the
regularised Boussinesq equations in the symmetric case. At the next order, the
system uncouples into two linear non-homogeneous ordinary differential
equations with variable coefficients, one correcting the localised part of the
solution, which we find analytically, and the other describing the
co-propagating oscillatory tail. This latter equation is a fourth-order
ordinary differential equation and is solved approximately by two different
methods, each exploiting the assumption that the leading solitary wave has a
small amplitude, and thus enabling an explicit estimate for the amplitude of
the oscillating tail. These estimates are compared with corresponding numerical
simulations.
