Remarks on certain two-component systems with peakon solutions

Hay Mike; Hone Andrew N. W.; Novikov Vladimir S.; Wang Jing Ping

doi:10.3934/jgm.2019028

Article Contents

2019, Volume 11, Issue 4: 561-573. Doi: 10.3934/jgm.2019028

This issue Previous Article Navier-Stokes and stochastic Navier-Stokes equations via Lagrange multipliers Next Article Non-Abelian momentum polytopes for products of

$ \mathbb{CP}^2 $

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

This article is dedicated to Darryl Holm on his 70th birthday.

Received: April 2018

Revised: June 2019

Published: November 2019

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

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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.

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Mathematics Subject Classification: Primary: 35Q51, 35G50; Secondary: 35Q53.

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