A connection between HH3 and KdV with one source

TL;DR

This paper explores the integrability of a KdV system with a source, revealing conditions for potential components, introducing a new potential term, and linking it to the He'non-Heiles Hamiltonian system through traveling wave reductions.

Contribution

It demonstrates the necessity of equal potentials in the source components, introduces an additional potential term, and establishes a connection to the integrable He'non-Heiles system.

Findings

Source components must have the same potential.

Introduction of an additional potential term preserves integrability.

Traveling wave reduction links to a case of the He'non-Heiles system.

Abstract

In the system made of Korteweg-de Vries with one source, we first show by applying the Painleve' test that the two components of the source must have the same potential. We then explain the natural introduction of an additional term in the potential of the source equations while preserving the existence of a Lax pair. This allows us to prove the identity between the travelling wave reduction and one of the three integrable cases of the cubic He'non-Heiles Hamiltonian system.