Two-Parameter Generalizations of Cauchy Bi-Orthogonal Polynomials and Integrable Lattices

Chang, Xiang-Ke; Li, Shihao; Tsujimoto, Satoshi; Yu, Guo‐Fu

doi:10.1007/s00332-021-09690-9

Cited by 5 publications

(4 citation statements)

References 21 publications

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“…Recall that it was shown in [15,18] that one-parameter deformations on the measures for nonsymmetric CBOPs and PSOPs induce integrable lattices. And, as mentioned in the above subsection, the Novikov peakon lattice describes an isospectral deformation associated with the discrete dual cubic string.…”

Section: Inverse Problem Given the Spectral Data {ζmentioning

confidence: 99%

“…The role of PSOPs is highlighted in the approximation problems germane to the inverse problem. (iii) Internal structures of Hermite-Padé approximation problems involved in the discrete dual cubic string related to Novikov multipeakons are investigated, as a result of which, intrinsic relations between a family of specific nonsymmetric CBOPs and PSOPs are obtained together with underlying relationships among the corresponding integrable lattices including the Novikov peakon lattice, B-Toda lattice [17] and an integrable lattice proposed in [18].…”

mentioning

confidence: 99%

“…5.4.3.A connection between two integrable lattices. The recurrence relation for nonsymmetric CBOPs together with one isospectral deformation was presented in[18, Proposition 3.4], from which we obtain that two sequences of monic nonsymmetric CBOPs {P C k (z)} and {Q C…”

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confidence: 99%

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Hermite--Padé approximations with Pfaffian structures: Novikov peakon equation and integrable lattices

Chang¹

2021

Preprint

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Motivated by the Novikov equation and its peakon problem, we propose a new mixed type Hermite-Padé approximation whose unique solution is a sequence of polynomials constructed with the help of Pfaffians. These polynomials belong to the family of recently proposed partial-skew-orthogonal polynomials. The relevance of partialskew-orthogonal polynomials is especially visible in the approximation problem germane to the Novikov peakon problem so that we obtain explicit inverse formulae in terms of Pfaffians by reformulating the inverse spectral problem for the Novikov multipeakons.Furthermore, we investigate two Hermite-Padé approximations for the related spectral problem of the discrete dual cubic string, and show that these approximation problems can also be solved in terms of partial-skew-orthogonal polynomials and nonsymmetric Cauchy biorthogonal polynomials. This formulation results in a new correspondence among several integrable lattices.

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Section: Inverse Problem Given the Spectral Data {ζmentioning

confidence: 99%

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confidence: 99%

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Hermite--Padé approximations with Pfaffian structures: Novikov peakon equation and integrable lattices

Chang¹

2021

Preprint

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“…During the submission of this paper, the spectral problem given in the studies of the Frobenius manifold and extended affine Weyl group W(k,k+1) (A l ) (5.1) has been connected with a two-parameter generalisation of the Cauchy two-matrix model. For more details, see[18].…”

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confidence: 99%

Integrable lattice hierarchies behind Cauchy two-matrix model and Bures ensemble

2022

Nonlinearity

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This paper focuses on different reductions of the two-dimensional (2d)-Toda hierarchy. Symmetric and skew-symmetric moment matrices are first considered, resulting in differential relations between symmetric/skew-symmetric tau functions and 2d-Toda’s tau functions, respectively. Furthermore, motivated by the Cauchy two-matrix model and Bures ensemble from random matrix theory, we study the rank-one shift condition in the symmetric case and rank-two shift condition in the skew-symmetric case, from which the C-Toda and B-Toda hierarchies are determined, together with their special Lax matrices and integrable structures.

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Hermite–Padé approximations with Pfaffian structures: Novikov peakon equation and integrable lattices

Chang

2022

Advances in Mathematics

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Two-Parameter Generalizations of Cauchy Bi-Orthogonal Polynomials and Integrable Lattices

Cited by 5 publications

References 21 publications

Hermite--Padé approximations with Pfaffian structures: Novikov peakon equation and integrable lattices

Hermite--Padé approximations with Pfaffian structures: Novikov peakon equation and integrable lattices

Integrable lattice hierarchies behind Cauchy two-matrix model and Bures ensemble

Hermite–Padé approximations with Pfaffian structures: Novikov peakon equation and integrable lattices

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