Related papers: Modulation of Camassa--Holm equation and reciproca…
The $r$-KdV-CH hierarchy is a generalization of the Korteweg-de Vries and Camassa-Holm hierarchies parametrized by $r+1$ constants. In this paper we clarify some properties of its multi-Hamiltonian structures, prove the semisimplicity of…
A reciprocal transformation for a 3-component Camassa-Holm type system is constructed to connect it with the first negative flow of a generalized MKdV hierarchy, and a bi-Hamiltonian structure for the transformed system is also considered.
We consider hydrodynamic systems which possess a local Hamiltonian structure of Dubrovin-Novikov type. To such a system there are also associated an infinite number of nonlocal Hamiltonian structures. We give necessary and sufficient…
A coupled Camassa-Holm type equation is linked to the first negative flow of a modified Drinfeld-Sokolov III hierarchy by a transformation of reciprocal type. Meanwhile the Lax pair and bi-Hamiltonian structure behaviors of this coupled…
A deformation parameter of a bihamiltonian structure of hydrodynamic type is shown to parameterize different extensions of the AKNS hierarchy to include negative flows. This construction establishes a purely algebraic link between, on the…
Original Whitham's method of derivation of modulation equations is applied to systems whose dynamics is described by a perturbed Korteweg-de Vries equation. Two situations are distinguished: (i) the perturbation leads to appearance of…
We discuss a reciprocal transformation for a three-component Camassa-Holm type equation and find that the transformed system is a reduction of the first negative flow for an extended MKdV hierarchy.
An explicit reciprocal transformation between a 2-component generalization of the Camassa-Holm equation, called the 2-CH system, and the first negative flow of the AKNS hierarchy is established, this transformation enables one to obtain…
The supersymmetric analog of the reciprocal transformation is introduced. This is used to establish a transformation between one of the supersymmetric Harry Dym equations and the supersymmetric modified Korteweg-de Vries equation. The…
In the present paper, a two-component Camassa-Holm (2CH) system with vorticity is studied as a geodesic flow on a suitable Lie group. The paper aims at presenting various details of the geometric formalism and a major result is the…
The bi-Hamiltonian structure is established for the perturbation equations of KdV hierarchy and thus the perturbation equations themselves provide also examples among typical soliton equations. Besides, a more general bi-Hamiltonian…
We derive the 2-component Camassa-Holm equation and corresponding N=1 super generalization as geodesic flows with respect to the $H^1$ metric on the extended Bott-Virasoro and superconformal groups, respectively.
The Camassa-Holm (CH) and Hunter-Saxton (HS) equations have an interpretation as geodesic flow equations on the group of diffeomorphisms, preserving the $H^1$ and $\dot{H}^1$ right-invariant metrics correspondingly. There is an analogy to…
This paper examines a generalization of the Camassa-Holm equation from the perspective of integrability. Using the framework developed by Dubrovin on bi-Hamiltonian deformations and the general theory of quasi-integrability, we demonstrate…
We sketch out a new geometric framework to construct Hamiltonian operators for generic, non-evolutionary partial differential equations. Examples on how the formalism works are provided for the KdV equation, Camassa-Holm equation, and…
We develop a systematic procedure for constructing soliton solutions of an integrable two-component Camassa-Holm (CH2) system. The parametric representation of the multisoliton solutions is obtained by using a direct method combined with a…
We introduce a bi-Hamiltonian hierarchy on the loop-algebra of sl(2) endowed with a suitable Poisson pair. It gives rise to the usual CH hierarchy by means of a bi-Hamiltonian reduction, and its first nontrivial flow provides a 3-component…
We provide a detailed treatment of the Camassa--Holm (CH) hierarchy with special emphasis on its algebro-geometric solutions. In analogy to other completely integrable hierarchies of soliton equations such as the KdV or AKNS hierarchies,…
In this paper, we provide the geometric formulation to the two-component Camassa-Holm equation (2-mCHE). We also study the relation between the 2-mCHE and the M-CV equation. We have shown that these equations arise from the invariant space…
We show that the bi-Hamiltonian structures of the Camassa-Holm and Harry Dym hierarchies can be obtained by applying a reduction process to a simple Poisson pair defined on the loop algebra of $\mathfrak{sl}(2,\mathbb{R})$. The reduction…