A connection between HH3 and KdV with one source

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.

💡 Research Summary

The paper investigates a coupled system consisting of the Korteweg‑de Vries (KdV) equation together with a single “source” consisting of two scalar fields. The authors first apply the Painlevé test to the full set of partial differential equations. By constructing Laurent expansions for each dependent variable and examining the resonances, they find that the compatibility conditions can only be satisfied if the two source components share exactly the same potential function. In other words, the derivatives of the potentials must coincide, which forces the two fields to be governed by an identical nonlinear term. This result eliminates the possibility of independent potentials for the two source components if one wishes to retain the Painlevé property, a hallmark of integrable systems.

Having established the necessity of a common potential, the authors turn to the Lax representation. They start from the standard scalar Lax pair of the KdV equation, (L=\partial_x^2+u) and (M=-4\partial_x^3-6u\partial_x-3u_x), and embed the source fields into a 2×2 matrix extension of (L). The extended operator includes off‑diagonal entries proportional to the source field (\phi) and a new quadratic term (\gamma\phi^2) in the diagonal. By demanding the zero‑curvature condition (