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

Chang, Xiang-Ke; Hu, Xing−Biao; Li, Shihao; Zhao, Jinping

doi:10.1016/j.aim.2018.09.023

Cited by 11 publications

(24 citation statements)

References 44 publications

(76 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The following conclusion is a consequence of combining Th. 3.7, 3.9 and 3.10 in [12], which implies that it is reasonable to view the B-Toda lattice and Novikov peakon lattice as opposite flows. Just note that here we establish a straightforward connection between the B-Toda lattice (instead of the modified B-Toda lattice) and Novikov peakon lattice.…”

Section: Appendix B Novikov Peakon and Finite B-toda Latticesmentioning

confidence: 99%

“…Recall that the CH peakon lattice is intimately linked to the finite A-Toda lattice 2 [3], while the Novikov peakon lattice is related to the finite B-Toda lattice [12]. This section is devoted to deriving the similar relevance for the DP peakon lattice and the associated C-Toda lattice.…”

Section: Dp Peakon and C-toda Latticesmentioning

confidence: 99%

“…Clearly, our proof implies that the DP peakon lattice can be seen as an isospectral flow on a manifold cut out by determinant identities. Recall that the CH peakon lattice is a flow on a manifold cut out by determinant identities [10], while the Novikov peakon lattice corresponds to a manifold cut out by Pfaffian identities [12]. The difference arises from the different structures of the corresponding tau functions belonging to different Lie algebra of A,B,C types.…”

Section: 4mentioning

confidence: 99%

“…The correspondence of the Novikov peakon and finite B-Toda lattices was investigated in the recent work [12]. This appendix is devoted to listing the main bridge between them with a slight modification.…”

Section: Appendix B Novikov Peakon and Finite B-toda Latticesmentioning

confidence: 99%

“…The answer is positive. In [12], the result for the Novikov peakon case is reported. It is shown that the Novikov peakon dynamical system is connected with the finite Toda lattice of BKP type (B-Toda lattice) and their solutions are related to the partition function of Bures ensemble with discrete measure (The readers are invited to see Appendix B for a short summary).…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

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

Chang

2018

Nonlinearity

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Appendix B Novikov Peakon and Finite B-toda Latticesmentioning

confidence: 99%

Section: Dp Peakon and C-toda Latticesmentioning

confidence: 99%

Section: 4mentioning

confidence: 99%

Section: Appendix B Novikov Peakon and Finite B-toda Latticesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Chang

2018

Nonlinearity

Self Cite

View full text Add to dashboard Cite

show abstract

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

Chang

et al. 2018

Commun. Math. Phys.

Self Cite

View full text Add to dashboard Cite

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] .

show abstract

Isospectral Flows Related to Frobenius–Stickelberger–Thiele Polynomials

Chang¹,

Hu²,

Szmigielski³

et al. 2019

Commun. Math. Phys.

Self Cite

View full text Add to dashboard Cite

The isospectral deformations of the Frobenius-Stickelberger-Thiele (FST) polynomials introduced in [32] (Spiridonov et al. Commun. Math. Phys. 272:139-165, 2007 ) are studied. For a specific choice of the deformation of the spectral measure, one is led to an integrable lattice (FST lattice), which is indeed an isospectral flow connected with a generalized eigenvalue problem. In the second part of the paper the spectral problem used previously in the study of the modified Camassa-Holm (mCH) peakon lattice is interpreted in terms of the FST polynomials together with the associated FST polynomials, resulting in a map from the mCH peakon lattice to a negative flow of the finite FST lattice. Furthermore, it is pointed out that the degenerate case of the finite FST lattice unexpectedly maps to the interlacing peakon ODE system associated with the two-component mCH equation studied in [17] (Chang et al. Adv. Math. 299:1-35, 2016).

show abstract

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

Cited by 11 publications

References 44 publications

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

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

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

Isospectral Flows Related to Frobenius–Stickelberger–Thiele Polynomials

Contact Info

Product

Resources

About