  We examine the duality relating the equilibrium dynamics of the mean-field
$p$-spin ferromagnets at finite size in the Guerra's interpolation scheme and
the Burgers hierarchy. In particular, we prove that - for fixed $p$ - the
expectation value of the order parameter on the first side w.r.t. the
generalized partition function satisfies the $p-1$-th element in the
aforementioned class of nonlinear equations. In the light of this duality, we
interpret the phase transitions in the thermodynamic limit of the statistical
mechanics model with the development of shock waves in the PDE side. We also
obtain the solutions for the $p$-spin ferromagnets at fixed $N$, allowing us to
easily generate specific solutions of the corresponding equation in the Burgers
hierarchy. Finally, we obtain an effective description of the finite $N$
equilibrium dynamics of the $p=2$ model with some standard tools in PDE side.
