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Time-Dependent Polynomials with One Double Root, and Related New Solvable Systems of Nonlinear Evolution Equations

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Abstract

Recently new solvable systems of nonlinear evolution equations—including ODEs, PDEs and systems with discrete time—have been introduced. These findings are based on certain convenient formulas expressing the kth time-derivative of a root of a time-dependent monic polynomial in terms of the kth time-derivative of the coefficients of the same polynomial and of the roots of the same polynomial as well as their time-derivatives of order less than k. These findings were restricted to the case of generic polynomials without any multiple root. In this paper some of these findings—those for \(k=1\) and \(k=2\)—are extended to polynomials featuring one double root; and a few representative examples are reported of new solvable systems of nonlinear evolution equations.

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Authors and Affiliations

  1. Department of Mathematics, University of Colorado, Colorado Springs, 1420 Austin Bluffs Pkway, Colorado Springs, CO, 80918, USA

    Oksana Bihun

  2. Physics Department, University of Rome “La Sapienza”, Rome, Italy

    Francesco Calogero

  3. Istituto Nazionale di Fisica Nucleare, Sezione di Roma, Rome, Italy

    Francesco Calogero

Authors
  1. Oksana Bihun
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  2. Francesco Calogero
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Correspondence to Oksana Bihun.

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Bihun, O., Calogero, F. Time-Dependent Polynomials with One Double Root, and Related New Solvable Systems of Nonlinear Evolution Equations. Qual. Theory Dyn. Syst. 18, 153–181 (2019). https://doi.org/10.1007/s12346-018-0282-3

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  • DOI: https://doi.org/10.1007/s12346-018-0282-3

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