# Hamiltonian Structures and Reciprocal Transformations for the r-KdV-CH Hierarchy

@article{Chen2008HamiltonianSA, title={Hamiltonian Structures and Reciprocal Transformations for the r-KdV-CH Hierarchy}, author={Ming Chen and Si‐Qi Liu and You-jin Zhang}, journal={arXiv: Exactly Solvable and Integrable Systems}, year={2008} }

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 the associated bihamiltonian structures and the formula for their central invariants. By introducing a class of generalized Hamiltonian structures, we give in a natural way the transformation formulae of the Hamiltonian structures of the hierarchy under certain…

## One Citation

### Multi-component generalizations of the CH equation: geometrical aspects, peakons and numerical examples

- Mathematics
- 2010

The Lax pair formulation of the two-component Camassa-Holm equation (CH2) is generalized to produce an integrable multi-component family, CH(n,k), of equations with n components and n velocities, which shows fluid-dynamics properties with coherent solitons following particle characteristics.

## References

SHOWING 1-10 OF 40 REFERENCES

### Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov - Witten invariants

- Mathematics
- 2001

We present a project of classification of a certain class of bihamiltonian 1+1 PDEs depending on a small parameter. Our aim is to embed the theory of Gromov - Witten invariants of all genera into the…

### Coupled Harry Dym equations with multi-Hamiltonian structures

- Physics
- 1988

The authors consider the isospectral flows of ( delta 2+ Sigma 1N upsilon i lambda i) Psi = alpha Psi . Using an unusual form of the 'Lax approach' they derive in a particularly simple manner: (a)…

### Bihamiltonian Hierarchies in 2D Topological Field Theory At One-Loop Approximation

- Mathematics
- 1998

Abstract:We compute the genus one correction to the integrable hierarchy describing coupling to gravity of a 2D topological field theory. The bihamiltonian structure of the hierarchy is given by a…

### Reciprocal transformations of Hamiltonian operators of hydrodynamic type: nonlocal Hamiltonian formalism for linearly degenerate systems

- Mathematics
- 2003

Reciprocal transformations of Hamiltonian operators of hydrodynamic type are investigated. The transformed operators are generally nonlocal, possessing a number of remarkable algebraic and…

### Integrable Systems and Classification of 2-dimensional Topological Field Theories

- Mathematics
- 1992

. In this paper we consider from the point of view of diﬀerential geometry and of the theory of integrable systems the so-called WDVV equations as deﬁning relations of 2-dimensional topological ﬁeld…

### A Darboux theorem for Hamiltonian operators in the formal calculus of variations

- Mathematics
- 2000

We prove a Darboux theorem for formal deformations of Hamiltonian operators of hydrodynamic type (Dubrovin-Novikov). Not all deformations are equivalent to the original operator: there is a moduli…

### Integrable Systems and Classification of 2-Dimensional Topological Field Theories

- Mathematics
- 1993

In this paper we consider the so-called WDVV equations from the point of view of differential geometry and of the theory of integrable systems as defining relations of 2-dimensional topological field…

### Frobenius Manifolds and Central Invariants for the Drinfeld - Sokolov Bihamiltonian Structures

- Mathematics
- 2007

### A New Integrable Equation with Peakon Solutions

- Mathematics
- 2002

We consider a new partial differential equation recently obtained by Degasperis and Procesi using the method of asymptotic integrability; this equation has a form similar to the Camassa–Holm shallow…