American Institute of Mathematical Sciences

December  2019, 11(4): 561-573. doi: 10.3934/jgm.2019028

Remarks on certain two-component systems with peakon solutions

 1 Dipartimento di Matematica e Fisica, Università di Roma Tre, 00146 Roma RM, Italy 2 School of Mathematics & Statistics, University of New South Wales, NSW 2052 Sydney, Australia 3 School of Mathematics, Loughborough University, Loughborough LE11 3TU, UK 4 School of Mathematics, Statistics & Actuarial Science, University of Kent, Canterbury CT2 7FS, UK

* Corresponding author: Andrew Hone; permanent address: SMSAS, University of Kent, Canterbury, UK

Received  April 2018 Revised  June 2019 Published  November 2019

Fund Project: The second author is supported by EPSRC fellowship EP/M004333/1. The fourth author is supported by EPSRC grant EP/P012698/1.

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.

Citation: Mike Hay, Andrew N. W. Hone, Vladimir S. Novikov, Jing Ping Wang. Remarks on certain two-component systems with peakon solutions. Journal of Geometric Mechanics, 2019, 11 (4) : 561-573. doi: 10.3934/jgm.2019028
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