  We consider the Adlam-Allen (AA) system of partial differential equations
which, arguably, is the first model that was introduced to describe solitary
waves in the context of propagation of hydrodynamic disturbances in
collisionless plasmas. Here, we identify the solitary waves of the model by
implementing a dynamical systems approach. The latter suggests that the model
also possesses periodic wave solutions --which reduce to the solitary wave in
the limiting case of infinite period-- as well as rational solutions which are
obtained herein. In addition, employing a long-wave approximation via a
relevant multiscale expansion method, we establish the asymptotic reduction of
the AA system to the Korteweg-de Vries equation. Such a reduction, is not only
another justification for the above solitary wave dynamics, but also may offer
additional insights for the emergence of other possible plasma waves. Direct
numerical simulations are performed for the study of multiple solitary waves
and their pairwise interactions. The stability of solitary waves is discussed
in terms of potentially relevant criteria, while the robustness of spatially
periodic wave solutions is touched upon by our numerical experiments.
