Time-dependent polynomials with one double root, and related new solvable systems of nonlinear evolution equations

TL;DR

This paper extends recent methods for solving nonlinear evolution equations by including polynomials with a double root, leading to new solvable systems and explicit formulas for derivatives of roots.

Contribution

It generalizes existing formulas for derivatives of polynomial roots to include polynomials with one double root, enabling the derivation of new solvable nonlinear evolution systems.

Findings

Extended formulas for first and second derivatives of roots with a double root

Derived new solvable nonlinear evolution equations involving double roots

Provided examples illustrating the application of the extended formulas

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 $k$-th time-derivative of a root of a time-dependent monic polynomial in terms of the $k$-th 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.