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Hermite-Padé approximations with Pfaffian structures: Novikov peakon equation and integrable lattices. (English) Zbl 1507.41007

Author’s abstract: 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.

MSC:

41A21 Padé approximation
30E10 Approximation in the complex plane
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K60 Lattice dynamics; integrable lattice equations
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
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