Nonlinear Systems and Their Remarkable Mathematical Structures: Volume I, 1st Edition (Hardback) book cover

Nonlinear Systems and Their Remarkable Mathematical Structures

Volume I, 1st Edition

Edited by Norbert Euler

CRC Press

582 pages | 11 Color Illus. | 26 B/W Illus.

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Hardback: 9781138601000
pub: 2018-12-03
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Nonlinear Systems and Their Remarkable Mathematical Structures aims to describe the recent progress in nonlinear differential equations and nonlinear dynamical systems (both continuous and discrete). Written by experts, each chapter is self-contained and aims to clearly illustrate some of the mathematical theories of nonlinear systems. The book should be suitable for some graduate and postgraduate students in mathematics, the natural sciences, and engineering sciences, as well as for researchers (both pure and applied) interested in nonlinear systems. The common theme throughout the book is on solvable and integrable nonlinear systems of equations and methods/theories that can be applied to analyze those systems. Some applications are also discussed.


  • Collects contributions on recent advances in the subject of nonlinear systems
  • Aims to make the advanced mathematical methods accessible to the non-expert in this field
  • Written to be accessible to some graduate and postgraduate students in mathematics and applied mathematics
  • Serves as a literature source in nonlinear systems


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

Table of Contents

Part A: Nonlinear Integrable Systems

A1. Systems of nonlinearly-coupled differential equations solvable

F Calogero

A2. Integrable nonlinear PDEs on the half-line

A S Fokas and B Pelloni

A3. Detecting discrete integrability: the singularity approach

Grammaticos, A Ramani, R Willox and T Mase

A4. Elementary introduction to discrete soliton equations J Hietarinta

A5. New results on integrability of the Kahan-Hirota-Kimura discretizations

Yu B Suris and M Petrera

Part B: Solution Methods and Solution Structures

B1. Dynamical systems satisfied by special polynomials and related isospectral matrices defined in terms of their zeros

O Bihun

B2. Singularity methods for meromorphic solutions of differential equations

R Conte, T W Ng and C Wu

B3. Pfeiffer-Sato solutions of Buhl's problem and a Lagrange-D'Alembert principle for Heavenly equations

O E Hentosh, Ya A Prykarpatsky, D Blackmore and A Prykarpatski

B4. Superposition formulae for nonlinear integrable equations in bilinear form

X B Hu

B5. Matrix solutions for equations of the AKNS system

C Schiebold

B6. Algebraic traveling waves for the generalized KdV-Burgers equation and the Kuramoto-Sivashinsky equation

C Valls

Part C: Symmetry Methods for Nonlinear Systems

C1. Nonlocal invariance of the multipotentialisations of the Kupershmidt equation and its higher-order hierarchies

M Euler and N Euler

C2. Geometry of normal forms for dynamical systems

G Gaeta

C3. Computing symmetries and recursion operators of evolutionary super-systems using the SsTools environment

A V Kiselev, A O Krutov and T Wolf

C4. Symmetries of It^o stochastic differential equations and their applications R Kozlov

C5. Statistical symmetries of turbulence

M Oberlack, M Wac lawczyk and V Grebenev

Part D: Nonlinear Systems in Applications

D1. Integral transforms and ordinary differential equations of infinite order

A Chavez, H Prado and E G Reyes

D2. The role of nonlinearity in geostrophic ocean flows on a sphere

A Constantin and R S Johnson

D3. Review of results on a system of type many predators - one prey

A V Osipov and G Soderbacka

D4. Ermakov-type systems in nonlinear physics and continuum mechanics

C Rogers and W K Schief

About the Editor

Norbert Euler is a professor of mathematics at Luleå University of Technology in Sweden. He is teaching a wide variety of mathematics courses at both the undergraduate and postgraduate level and has done so at several universities worldwide for more than 25 years. He is an active researcher and has to date published over 70 peer reviewed research articles in the subject of nonlinear systems and he is a co-author of several books. He is also involved in editorial work for some international journals, and he is the Editor-in-Chief of the Journal of Nonlinear Mathematical Physics since 1997.

Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Arithmetic
MATHEMATICS / Differential Equations