This equation exhibits an infinite number of nontrivial
conserved
quantities and has analytic solutions in spite on being
nonlinear.
Later, it was noticed that this was only a particular
equation in an
infinite set of integrable systems, a process which culminated
in the
work of Drinfeld and Sokolov, where a classification
scheme
based in a Lax-type formalism was constructed. Recently,
integrable
systems have been shown to be related to Conformal Field
Theory,
and this has sparkled a new interest in the field.
Here we review the KdV-type hierarchies of equations
using the pseudodifferential operator (PDO) formalism
of Gelfand and
Dickey. We emphasize the bihamiltonian structure which
seems to be the
fundamental mark of integrable systems. Our presentation
closely follows
Dickey, L.A., Soliton equations and hamiltonian
systems, Advanced
Series in Mathematical Physics
Vol. 12, World Scientific (1991).
Technical Report (40 pages)