Hermite--Pad\'{e} approximations with Pfaffian structures: Novikov peakon equation and integrable lattices

TL;DR

This paper introduces a novel Hermite--Padé approximation method using Pfaffians, linking Novikov peakon equations, partial-skew-orthogonal polynomials, and integrable lattices, with explicit inverse formulas and spectral problem solutions.

Contribution

It develops a new mixed Hermite--Padé approximation framework involving Pfaffians, connecting peakon problems, spectral theory, and integrable lattice structures.

Findings

Explicit inverse formulas in terms of Pfaffians for Novikov peakon inverse spectral problem

Solution of Hermite--Padé approximation problems using partial-skew-orthogonal and Cauchy biorthogonal polynomials

Establishment of a new correspondence among integrable lattices

Abstract

Motivated by the Novikov equation and its peakon problem, we propose a new mixed type Hermite--Pad\'{e} 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 partial-skew-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--Pad\'{e} 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.