Isospectral flows related to Frobenius-Stickelberger-Thiele polynomials. (English) Zbl 1445.37052
Summary: The isospectral deformations of the Frobenius-Stickelberger-Thiele (FST) polynomials introduced in [V. P. Spiridonov et al., Commun. Math. Phys. 272, No. 1, 139–165 (2007; Zbl 1136.37041)] are studied. For a specific choice of the deformation of the spectral measure, one is led to an integrable lattice (FST lattice), which is indeed an isospectral flow connected with a generalized eigenvalue problem. In the second part of the paper the spectral problem used previously in the study of the modified Camassa-Holm (mCH) peakon lattice is interpreted in terms of the FST polynomials together with the associated FST polynomials, resulting in a map from the mCH peakon lattice to a negative flow of the finite FST lattice. Furthermore, it is pointed out that the degenerate case of the finite FST lattice unexpectedly maps to the interlacing peakon ODE system associated with the two-component mCH equation studied in [X.-K. Chang et al., Adv. Math. 299, 1–35 (2016; Zbl 1353.37139)].
MSC:
| 37K60 | Lattice dynamics; integrable lattice equations |
| 33E05 | Elliptic functions and integrals |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 45C05 | Eigenvalue problems for integral equations |
| 34A55 | Inverse problems involving ordinary differential equations |
| 35Q51 | Soliton equations |
Keywords:
isospectral deformations; Frobenius-Stickelberger-Thiele polynomials; Camassa-Holm peakon latticeReferences:
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