Zeros of Polynomials and Solvable Nonlinear Evolution Equations

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.

Francesco Calogero is Professor of Theoretical Physics (Emeritus), at the Università degli Studi di Roma 'La Sapienza', Italyof Rome, Italy. He has published numerous papers and books on physics and mathematics as well as on arms control and disarmament topics and he served as Secretary General of the Pugwash Conferences on Science and World Affairs which won the 1995 Nobel Peace Prize.