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Isospectral Flows Related to Frobenius–Stickelberger–Thiele Polynomials

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Abstract

The isospectral deformations of the Frobenius–Stickelberger–Thiele (FST) polynomials introduced in Spiridonov et al. (Commun Math Phys 272:139–165, 2007) 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 Chang et al. (Adv Math 299:1–35, 2016).

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Acknowledgements

X.C. was supported in part by the National Natural Science Foundation of China (Grant Nos. 11688101, 11731014, 11701550) and the Youth Innovation Promotion Association CAS. X.H. was supported in part by the National Natural Science Foundation of China (Grant Nos. 11931017 and 11871336). J.S. was supported in part by the Natural Sciences and Engineering Research Council of Canada. A.Z. was supported in part by the National Natural Science Foundation of China (Grant No. 11771015).

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Authors and Affiliations

  1. LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, P.O. Box 2719, Beijing, 100190, People’s Republic of China

    Xiang-Ke Chang & Xing-Biao Hu

  2. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, People’s Republic of China

    Xiang-Ke Chang & Xing-Biao Hu

  3. Department of Mathematics and Statistics, University of Saskatchewan, 106 Wiggins Road, Saskatoon, Saskatchewan, S7N 5E6, Canada

    Jacek Szmigielski

  4. School of Mathematics, Renmin University of China, Beijing, 100872, China

    Alexei Zhedanov

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  1. Xiang-Ke Chang
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  2. Xing-Biao Hu
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  4. Alexei Zhedanov
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Correspondence to Xiang-Ke Chang.

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Communicated by P. Deift

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Chang, XK., Hu, XB., Szmigielski, J. et al. Isospectral Flows Related to Frobenius–Stickelberger–Thiele Polynomials. Commun. Math. Phys. 377, 387–419 (2020). https://doi.org/10.1007/s00220-019-03616-z

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