Degasperis–Procesi peakon dynamical system and finite Toda lattice of CKP type

Abstract

In this paper, we propose a finite Toda lattice of CKP type (C-Toda) together with a Lax pair. Our motivation is based on the fact that the Camassa-Holm (CH) peakon dynamical system and the finite Toda lattice may be regarded as opposite flows in some sense. As an intriguing analogue to the CH equation, the Degasperis-Procesi (DP) equation also supports the presence of peakon solutions. Noticing that the peakon solution to the DP equation is expressed in terms of bimoment determinants related to the Cauchy kernel, we impose opposite time evolution on the moments and derive the corresponding bilinear equation. The corresponding quartic representation is shown to be a continuum limit of a discrete CKP equation, due to which we call the obtained equation finite Toda lattice of CKP type. Then, a nonlinear version of the C-Toda lattice together with a Lax pair is derived. As a result, it is shown that the DP peakon lattice and the finite C-Toda lattice form opposite flows under certain transformation.2010 Mathematics Subject Classification. 37K10, 35Q51, 15A15.

An application of Pfaffians to multipeakons of the Novikov equation and the finite Toda lattice of BKP type

“…The study of the CKP type is a bit different from that for the BKP type. We address the corresponding result elsewhere [13].…”

Frobenius manifolds and a new class of extended affine Weyl groups of A-type

“…• Is it possible to obtain an explicit realization of the integrable hierarchies associated with the Frobenius manifolds on the orbit space of W (k,k+1) (A l )? Perhaps this problem is related to the works in [23] or [24] about rational reductions of the 2D-Toda hierarchy, or in [25,26]…”

The Cauchy Two-Matrix Model, C-Toda Lattice and CKP Hierarchy

“…Here we would like to mention that we will give a proof of Proposition 3.1 by Pfaffian techniques in Appendix. A different proof by determinant techniques can be referred to [11].…”

“…It should be mentioned that the motivation of derivation C-Toda lattice doesn't only lie in the connection with sCBOPs, but in the study of the positive flow of Degasperis-Procesi peakon equation. A detailed discussion about the relationship between C-Toda lattice and Degasperis-Procesi equation can be referred to [11].…”