Integrable Quartic Potentials and Coupled KdV Equations

Abstract

We show a surprising connection between known integrable Hamiltonian systems with quartic potential and the stationary flows of some coupled KdV systems related to fourth order Lax operators. In particular, we present a connection between the Hirota-Satsuma coupled KdV system and (a generalisation of) the $1:6:1$ integrable case quartic potential. A generalisation of the $1:6:8$ case is similarly related to a different (but gauge related) fourth order Lax operator. We exploit this connection to derive a Lax representation for each of these integrable systems. In this context a canonical transformation is derived through a gauge transformation.