Nonlinear Systems and Their Remarkable Mathematical Structures

Reviews / Votes

The theory of integrable systems studies remarkable equations of mathematical physics which are, in a sense, exactly solvable and possess regular behaviour. Such equations play a fundamental role in applications by providing approximations to various (non-integrable) physical models. Dating back to Newton, Euler and Jacobi, the theory of integrable systems plays nowadays a unifying role in mathematics bringing together algebra, geometry and analysis.

This volume is a collection of invited contributions written by leading experts in the area of integrable dynamical systems and their applications. The content covers a wide range of topics, both classical and relatively recent. It provides a valuable source of information for both experts and the beginners. Various combinations of sections of the book would make excellent self-contained lecture courses.

This book will certainly be a valuable asset to any University library. Written by highly established and actively working researchers, it is quite unique in style due to the breath of the material covered. It will remain a valuable source of information for years to come.

-Evgeny Ferapontov, Loughborough University

The main purpose of the book Mathematica structures of nonlinear systems, first of a series, is to present the most recent and not widely known results on the mathematical tools necessary to construct solutions to nonlinear systems and their applications. All contributions present a long list of updated references which make the volume particularly useful also for beginners. The mathematical structures presented in this volume have universal applications in many fields of nature, a very limited number of which are presented in the final chapter. I found particularly interesting the presentations:

1. On the old problem of the integrability of nonlinear PDEs defined on half-lines by Fokas and Pelloni.

2. On the exact superposition formulae in bilinear form, ready for use, for the construction of sequences of exact solutions of many integrable nonlinear equations by Hu.

3. On the construction of nonlocal recursion operators for local symmetries of supersymmetric nonlinear equations by Kiselev, Krutov and Wolf.

4. On the role of nonlinearity in geostrophic ocean flow and its practical consequences by Constantin and Johnson.

-Decio Levi, Istituto Nazionale di Fisica Nucleare