Remarks on certain two-component systems with peakon solutions

TL;DR

This paper analyzes a family of two-component systems related to the Camassa-Holm equation, demonstrating how transformations simplify the system and exploring conditions for integrability, including non-integrability results for certain scalar equations.

Contribution

It shows that the apparent freedom in the Lax pair can be removed through transformations, reducing the system to a potentially integrable triangular form, and establishes non-integrability for higher-degree nonlinear Camassa-Holm type equations.

Findings

Transformations reduce the system to triangular form.

The integrability depends on the choice of the function H.

Scalar equations with nonlinear degree > 3 are not integrable.

Abstract

We consider a Lax pair found by Xia, Qiao and Zhou for a family of two-component analogues of the Camassa-Holm equation, including an arbitrary function $H$, and show that this apparent freedom can be removed via a combination of a reciprocal transformation and a gauge transformation, which reduces the system to triangular form. The resulting triangular system may or may not be integrable, depending on the choice of $H$. In addition, we apply the formal series approach of Dubrovin and Zhang to show that scalar equations of Camassa-Holm type with homogeneous nonlinear terms of degree greater than three are not integrable.