Two-parameter generalisations of Cauchy bi-orthogonal polynomials and integrable lattices

2020/07/12 by Xiang‐Ke Chang, Chang, Xiang-Ke, Shi‐Hao Li +5 · 1 citation
Mathematics · Physics and Astronomy · #15A15 #37K10 #37K20 #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Mathematical Analysis and Transform Methods #Mathematical Physics (math-ph) #Mathematical functions and polynomials #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.2007.05998

openalex publication_date 2020/07/12 · openalex created_date 2020/07/16 · openalex updated_date 2026/07/28

Abstract

In this article, we consider the generalised two-parameter Cauchy two-matrix model and corresponding integrable lattice equation. It is shown that with parameters chosen as 1/k_i when k_i∈ℤ_>0 (i=1, 2), the average characteristic polynomials admit (k₁+k₂+2)-term recurrence relations, which provide us spectral problems for integrable lattices. The tau function is then given by the partition function of the generalised Cauchy two-matrix model as well as Gram determinant. The simplest example with exact solvability is demonstrated.

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• Elliptic solutions of the lattice CKP equation and its elliptic direct linearisation scheme

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