  The long-time behaviour of solutions of systems of conservation laws has been
extensively studied. In particular, Liu and Zeng \cite{liu:1997} have given a
detailed exposition of the leading order asymptotics of solutions close to a
constant background state. In this paper, we extend the analysis of
\cite{liu:1997} by examining higher order terms in the asymptotics in the
framework of the so-called two dimensional p-system, though we believe that our
methods and results also apply to more general systems. We give a constructive
procedure for obtaining these terms, and we show that their structure is
determined by the interplay of the parabolic and hyperbolic parts of the
problem. In particular, we prove that the corresponding solutions develop long
tails that precede the characteristics.
