On the Camassa-Holm and K-dV Hierarchies
نویسندگان
چکیده
It is known that a transform of Liouville type allows one to pass from an equation of the Korteweg-de Vries (K-dV) hierarchy to a corresponding equation of the Camassa-Holm (CH) hierarchy [2, 39]. We give a systematic development of the correspondence between these hierarchies by using the coefficients of asymptotic expansions of certain Green’s functions. We illustrate our procedure with some examples.
منابع مشابه
On Time Fractional Modifed Camassa-Holm and Degasperis-Procesi Equations by Using the Haar Wavelet Iteration Method
The Haar wavelet collocation with iteration technique is applied for solving a class of time-fractional physical equations. The approximate solutions obtained by two dimensional Haar wavelet with iteration technique are compared with those obtained by analytical methods such as Adomian decomposition method (ADM) and variational iteration method (VIM). The results show that the present scheme is...
متن کاملOn the bi-Hamiltonian structures of the Camassa-Holm and Harry Dym equations
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 sl(2,R). The reduction process is a bi-Hamiltonian reduction, that can be canonically performed on every bi-Hamiltonian manifold.
متن کاملLiouville Correspondence Between the Modified KdV Hierarchy and Its Dual Integrable Hierarchy
We study an explicit correspondence between the integrable modified KdV hierarchy and its dual integrable modified Camassa-Holm hierarchy. A Liouville transformation between the isospectral problems of the two hierarchies also relates their respective recursion operators, and serves to establish the Liouville correspondence between their flows and Hamiltonian conservation laws. In addition, a n...
متن کامل1 0 Ju n 20 02 The Camassa - Holm hierarchy , N - dimensional integrable systems , r - matrix structure , and algebro - geometric solution
This paper shows that the Camassa-Holm (CH) spectral problem yields two different integrable hierarchies of nonlinear evolution equations (NLEEs), one is of negative order CH hierachy while the other one is of positive order CH hierarchy. The two CH hierarchies possess the zero curvature representations through solving a key matrix equation. We find that the well-known CH equation is included i...
متن کاملGeneralisations of the Camassa-Holm equation
We classify generalised Camassa-Holm type equations which possess infinite hierarchies of higher symmetries. We show that the obtained equations can be treated as negative flows of integrable quasi-linear scalar evolution equations of orders 2, 3 and 5. We present the corresponding Lax representations or linearisation transformations for these equations. Some of the obtained equations seem to b...
متن کامل