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.

翻译：基于诺维科夫方程式及其峰顶问题, 我们提出一种新的混合类型 Hermite- Pad\ {e} 近似, 其独特的解决方案是在Pfaffians的帮助下建造的多光谱序列。 这些多光谱属于最近提议的半skew- orthologal 多孔形多孔形多孔形的家族。 部分skew- orthogoal 多孔形多孔径多孔形的近端问题在与诺维科夫顶峰问题相关的近端问题中特别明显可见, 这样我们就能通过对诺维科夫多孔形多孔形的反光谱问题进行重新配置, 从而获得Pfafficatians 的明显反向公式。 此外, 我们调查了与离散双立双立线相关光谱问题的两种Hermite- Pad\ {e} 近光谱, 并表明这些近端问题也可以通过部分skew- orpotonocial多孔形多孔形多孔形多孔形的近端问题得到解决。 这种配方程式在几个不相间的新通信中产生新通信。