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.

　　Publication: 

　　Advances in Mathematics, Volume 402, 25 June 2022 

　　Author: 

　　Xiang-Ke Chang 

　　LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, P.O. Box 2719, Beijing 100190, PR China 

　　School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, PR China 

　　Email: changxk@lsec.cc.ac.cn