Nonlinear Systems and Their Remarkable Mathematical Structures Volume 1

Part A: Nonlinear Integrable Systems

A1. Systems of nonlinearly-coupled differential equations solvable

A2. Integrable nonlinear PDEs on the half-line

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

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

B4. Superposition formulae for nonlinear integrable equations in bilinear form

B5. Matrix solutions for equations of the AKNS system

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

Part C: Symmetry Methods for Nonlinear Systems

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

C2. Geometry of normal forms for dynamical systems

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

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

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

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

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

C Rogers and W K Schief

Biography

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.

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