Zeros of Polynomials and Solvable Nonlinear Evolution Equations

Exploring a novel breakthrough in the identification and investigation of solvable and integrable nonlinearly coupled evolution ordinary differential equations (ODEs) or partial differential equations (PDEs).

Francesco Calogero (Author)

9781108428590, Cambridge University Press

Hardback, published 20 September 2018

176 pages
25.3 x 18 x 1.3 cm, 0.53 kg

'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, MathSciNet

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.

Preface
1. Introduction
2. Parameter-dependent monic polynomials, definitions and key formulas
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
References.

Subject Areas: Mathematical physics [PHU], Statistical physics [PHS]