On an integrable multi-component Camassa–Holm system arising from Möbius geometry

Abstract

In this paper, we mainly study the geometric background, integrability and peaked solutions of a ( 1 + n ) -component Camassa–Holm (CH) system and some related multi-component integrable systems. Firstly, we show this system arises from the invariant curve flows in the Möbius geometry and serves as the dual integrable counterpart of a geometrical ( 1 + n ) -component Korteweg–de Vries system in the sense of tri-Hamiltonian duality. Moreover, we obtain an integrable two-component modified CH system using a generalized Miura transformation. Finally, we provide a necessary condition, under which the dual integrable systems can inherit the Bäcklund correspondence from the original ones.

Introduction to special feature dedicated to Prof. Allan Fordy on the occasion of his 70th birthday

“…A substantial part of Allan's work has been devoted to understanding the integrability of nonlinear evolution equations via their bi-Hamiltonian (or multi-Hamiltonian) structures [17]. The paper [18] by Kang and co-authors begins with hierarchies of KdV-type systems arising from curve flows in Möbius geometry, and then uses tri-Hamiltonian duality to obtain associated multi-component Camassa-Holm hierarchies. The contribution [19] of Pavlov et al on the other hand, is concerned with classifying bi-Hamiltonian pairs consisting of two operators of first order, one of which is perturned by a nonlocal term; this produces some interesting examples of 1 + 1-dimensional equations, namely the astigmatism equation, and a three-component system obtained by extending a solution of the Witten-Dijkgraaf-Verlinde-Verlinde equations.…”

“…A substantial part of Allan’s work has been devoted to understanding the integrability of nonlinear evolution equations via their bi-Hamiltonian (or multi-Hamiltonian) structures [17]. The paper [18] by Kang and co-authors begins with hierarchies of KdV-type systems arising from curve flows in Möbius geometry, and then uses tri-Hamiltonian duality to obtain associated multi-component Camassa-Holm hierarchies. The contribution [19] of Pavlov et al .…”