Zeros of Polynomials and Solvable Nonlinear Evolution Equations

Francesco Calogero

doi:10.1017/9781108553124

Reporting a novel breakthrough in the identification and investigation of solvable and integrable nonlinearly coupled evolution ordinary differential equations (ODEs) or partial differential equations (PDEs), this text includes practical examples throughout to illustrate the theoretical concepts. Beginning with systems of ODEs, including second-order ODEs of Newtonian type, it then discusses systems of PDEs, and systems evolving in discrete time. It reports a novel, differential algorithm which can be used to evaluate all the zeros of a generic polynomial of arbitrary degree: a remarkable development of a fundamental mathematical problem with a long history. The book will be of interest to applied mathematicians and mathematical physicists working in the area of integrable and solvable non-linear evolution equations; it can also be used as supplementary reading material for general applied mathematics or mathematical physics courses.

'This concise book discusses the mathematical tools used to model complex phenomena via systems of nonlinear equations, which can be useful to describe many-body problems. Starting from a well-established approach to solvable dynamical systems identification, the author proposes a novel algorithm that allows some of the restrictions of this approach to be eliminated and, thus, identifies more solvable/integrable N-body problems. … the book presents many examples to show its application and impact. … [It] is addressed to both applied mathematicians and theoretical physicists, and can be used as a basic text for a topical course for advanced undergraduates.'

Virginia Greco Source: CERN Courier

‘The monograph is self-contained and accessible to a wide range of readers. In particular, those interested in the use of transformation techniques, in the study of integrable systems, will find this book especially valuable and useful.’

D. Catalano Ferraioli Source: MathSciNet

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Zeros of Polynomials and Solvable Nonlinear Evolution Equations

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Contents

Frontmatter
pp i-iv

Contents
pp v-viii

Preface
pp ix-x

1 - Introduction
pp 1-3

2 - Parameter-Dependent Monic Polynomials: Definitions, Key Formulas and Other Preliminaries
pp 4-25

3 - A Differential Algorithm to Compute All the Zeros of a Generic Polynomial
pp 26-33

4 - Solvable and Integrable Nonlinear Dynamical Systems: Mainly Newtonian N-Body Problems in the Plane
pp 34-109

5 - Solvable Systems of Nonlinear Partial Differential Equations (PDEs)
pp 110-118

6 - Generations of Monic Polynomials
pp 119-142

7 - Discrete Time
pp 143-159

8 - Outlook
pp 160-161

Appendix - Complex Numbers and Real 2-Vectors
pp 162-163

References
pp 164-168

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