Integrable Quartic Potentials and Coupled KdV Equations

TL;DR

This paper uncovers a surprising link between integrable Hamiltonian systems with quartic potentials and stationary flows of coupled KdV systems, providing new Lax representations and transformations.

Contribution

It establishes a novel connection between quartic potential systems and coupled KdV equations, including generalizations and explicit Lax representations.

Findings

Connected quartic potentials to coupled KdV systems.

Derived Lax representations for these integrable systems.

Established gauge transformations and canonical transformations.

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.