DeepAI AI Chat
Log In Sign Up

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

by   Xiang-Ke Chang, et al.

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 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–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.


page 1

page 2

page 3

page 4


Orthogonal polynomials on planar cubic curves

Orthogonal polynomials in two variables on cubic curves are considered, ...

On generating Sobolev orthogonal polynomials

Sobolev orthogonal polynomials are polynomials orthogonal with respect t...

Gossip of Statistical Observations using Orthogonal Polynomials

Consider a network of agents connected by communication links, where eac...

Non-commutative Hermite–Padé approximation and integrability

We introduce and solve the non-commutative version of the Hermite-Padé t...

A Riemann–Hilbert approach to computing the inverse spectral map for measures supported on disjoint intervals

We develop a numerical method for computing with orthogonal polynomials ...

Continued fractions and orthogonal polynomials in several variables

We extend the close interplay between continued fractions, orthogonal po...

The conjugate gradient method with various viewpoints

Connections of the conjugate gradient (CG) method with other methods in ...