Soliton equations exhibiting Pfaffian solutions

2018

Nonlinearity

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.

Section: Conclusion and Discussionmentioning

confidence: 69%

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

2018

Nonlinearity

“…The equation (3.12) is a bilinear form of the finite B-Toda lattice. The bilinear form (3.12) has appeared as a special case of famous discrete BKP equation (also called Hirota-Miwa equation) [27,29,43]. In fact, from our derivation, we have given an explicit construction for the so-called "molecule solution" to the finite bilinear B-Toda lattice (3.12).…”

Section: )mentioning

confidence: 96%

“…Our motivation is that there has existed a generalized bilinear B-Toda lattice in [27,29]. Our goal was to investigate its molecule solution and we eventually succeeded.…”

Section: Generalizationsmentioning

confidence: 99%

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

Advances in Mathematics

et al. 2018

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.

“…which belongs to the discrete BKP hierarchy [23,30]. If we introduce the variables Next we show how the above results are related to the so-called generalized inverse vector-valued Padé approximant (GIPA) [24][25][26].…”

Section: Two Integrable Lattices As Algorithmsmentioning

confidence: 99%

“…which belongs to the discrete BKP hierarchy [23,30]. Note that this bilinear equation is a discrete analogue of (3.8), since (3.8) can be reproduced from (3.20) by taking appropriate limits with respect to the index l. In fact, by settinĝ…”

Section: )mentioning

confidence: 99%

Partial-Skew-Orthogonal Polynomials and Related Integrable Lattices with Pfaffian Tau-Functions

et al. 2018

Commun. Math. Phys.

Skew-orthogonal polynomials (SOPs) arise in the study of the n-point distribution function for orthogonal and symplectic random matrix ensembles. Motivated by the average of characteristic polynomials of the Bures random matrix ensemble studied in [22], we propose the concept of partial-skew-orthogonal polynomials (PSOPs) as a modification of the SOPs, and then the PSOPs with a variety of special skew-symmetric kernels and weight functions are addressed. By considering appropriate deformations of the weight functions, we derive nine integrable lattices in different dimensions. As a consequence, the tau-functions for these systems are shown to be expressed in terms of Pfaffians and the wave vectors PSOPs. In fact, the taufunctions also admit the representations of multiple integrals. Among these integrable lattices, some of them are known, while the others are novel to the best of our knowledge. In particular, one integrable lattice is related to the partition function of the Bures random matrix ensemble.Besides, we derive a discrete integrable lattice, which can be used to compute certain vector Padé approximants. This yields the first example regarding the connection between integrable lattices and vector Padé approximants, for which Hietarinta, Joshi and Nijhoff pointed out that " This field remains largely to be explored. " in the recent monograph [27, Section 4.4] .