Zeros of Polynomials and Solvable Nonlinear Evolution Equations

Francesco Calogero

• Publisher: Cambridge University Press
• Year: 2018
• Print ISBN: 9781108428590
• eBook ISBN: 9781108619080
• Topic: Physical Sciences
• Subtopic: Mathematical & Computational Physics

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.

Table of contents

• Cover
• Half-title
• Title page
• Copyright information
• Table of contents
• Preface
• 1 Introduction
• 2 Parameter-Dependent Monic Polynomials: Definitions, Key Formulas and Other Preliminaries
• 3 A Differential Algorithm to Compute All the Zeros of a Generic Polynomial
• 4 Solvable and Integrable Nonlinear Dynamical Systems: Mainly Newtonian N-Body Problems in the Plane
• 5 Solvable Systems of Nonlinear Partial Differential Equations (PDEs)
• 6 Generations of Monic Polynomials
• 7 Discrete Time
• 8 Outlook
• Appendix Complex Numbers and Real 2-Vectors
• References