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 ker… Show more
The Novikov equation is an integrable analogue of the Camassa-Holm equation with a cubic (rather than quadratic) nonlinear term. Both these equations support a special family of weak solutions called multipeakon solutions. In this paper, an approach involving Pfaffians is applied to study multipeakons of the Novikov equation. First, we show that the Novikov peakon ODEs describe an isospectral flow on the manifold cut out by certain Pfaffian identities. Then, a link between the Novikov peakons and the finite Toda lattice of BKP type (B-Toda lattice) is established based on the use of Pfaffians. Finally, certain generalizations of the Novikov equation and the finite B-Toda lattice are proposed together with special solutions. To our knowledge, it is the first time that the peakon problem is interpreted in terms of Pfaffians.2010 Mathematics Subject Classification. 37K10, 35Q51, 15A15.
The Novikov equation is an integrable analogue of the Camassa-Holm equation with a cubic (rather than quadratic) nonlinear term. Both these equations support a special family of weak solutions called multipeakon solutions. In this paper, an approach involving Pfaffians is applied to study multipeakons of the Novikov equation. First, we show that the Novikov peakon ODEs describe an isospectral flow on the manifold cut out by certain Pfaffian identities. Then, a link between the Novikov peakons and the finite Toda lattice of BKP type (B-Toda lattice) is established based on the use of Pfaffians. Finally, certain generalizations of the Novikov equation and the finite B-Toda lattice are proposed together with special solutions. To our knowledge, it is the first time that the peakon problem is interpreted in terms of Pfaffians.2010 Mathematics Subject Classification. 37K10, 35Q51, 15A15.
“…• 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]…”
We present a new class of extended affine Weyl groups W (k,k+1) (A l ) for 1 ≤ k < l and obtain an analogue of Chevalley-type theorem for their invariants. We further show the existence of Frobenius manifold structures on the orbit spaces of W (k,k+1) (A l ) and also construct Landau-Ginzburg superpotentials for these Frobenius manifold structures.
“…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].…”
Section: The T-deformations Of Scbops and The C-toda Latticementioning
confidence: 99%
“…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].…”
Section: The T-deformations Of Scbops and The C-toda Latticementioning
This paper mainly talks about the Cauchy two-matrix model and its corresponding integrable hierarchy with the help of orthogonal polynomials theory and Toda-type equations. Starting from the symmetric reduction of Cauchy biorthogonal polynomials, we derive the Toda equation of CKP type (or the C-Toda lattice) as well as its Lax pair by introducing time flows. Then, matrix integral solutions to the C-Toda lattice are extended to give solutions to the CKP hierarchy which reveals the time-dependent partition function of the Cauchy two-matrix model is nothing but the τ -function of the CKP hiearchy. At last, the connection between the Cauchy two-matrix model and Bures ensemble is established from the point of view of integrable systems.
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