Nonlinear Systems and Their Remarkable Mathematical Structures

Norbert Euler is professor of mathematics at Jinan University in Guangzhou, P. R. China, and visiting research professor at the Centro Internacional de Ciencias AC in Cuernavaca, Mexico. Until April 2019 he was professor of mathematics at Luleå University of Technology in Sweden, where he was teaching and researching for 23 years. His main research interests are in the subject of nonlinear mathematical physics, in particular nonlinear ordinary and partial differential equations and integrable systems, and he has published approximately 80 peer reviewed research articles and co-authored several books. He is involved in editorial work for journals, and has been the editor-in-chief of the Journal of Nonlinear Mathematical Physics since 1997. Maria Clara Nucci is associate professor of mathematical physics at University of Perugia, where she graduated in mathematics summa cum laude. Between 1986 and 1991 she was a visiting assistant professor at Georgia Institute of Technology, Atlanta, US. She has also been invited by universities in Australia, Canada, France, Germany, Greece, Sweden, UK, and the US. She has presented her research at many international congresses and workshops. From 1995–2009 she was associate editor of Journal of Mathematical Analysis and Applications, and since 2005 has been a member of the editorial board of Journal of Nonlinear Mathematical Physics. She is author or co-author of more than 100 publications, and has wide ranging research interests, from fluid to rigid body mechanics, epidemiology to astrophysics, and history of mathematics to quantum mechanics.

"This is an excellent complement to the first volume, with a roster of authors including many of the best contributors to the major developments on dynamical systems of the last few decades." —Francesco Calogero, Emeritus Professor, University of Rome "La Sapienza"“This book contains useful introductory and review articles on several topics connected with integrability. Among others it includes four articles on various symmetry approaches, and a nice introduction to cluster algebras and integrability. I particularly liked the extensive reviews of elliptic difference Painleve equations, and of Wronskian solutions of integrable systems. With so many good review articles, it will serve as a nice reference book for anybody working with integrable systems.”—Emeritus Professor Jarmo Hietarinta, University of Turku“This volume is a sequel to a recently launched project of publishing invited contributions written by leading experts in the area of integrable 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. The material is grouped into 3 Parts:Part A: Integrability, Lax pairs and Symmetry (reciprocal transformations, contact Lax pairs and dispersionless systems, Lax pairs for Boussinesq systems, symmetries of delay differential equations, symmetry approach to integrability, classical and non-classical symmetries). Part B: Algebraic and Geometric Methods (instabilities of NLS plane waves, rational solutions to Painleve equations, cluster algebras and discrete integrability, elliptic difference Painleve equations, linkage mechanisms and discrete curves, Cauchy problems and infinite-dimensional groups, Wronskian solutions). Part C: Applications (gradient catastrophe in shallow water model, vibrations of an elastic bar and modified Camassa-Holm equations, exactly solvable quantum mechanics and orthogonal polynomials). This book will certainly be a valuable asset to any University library. Written by 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.”—Professor E.V. Ferapontov, Loughborough University"This is the second in the pair of beautiful volumes casting a portrait of present day knowledge of nonlinear systems, with contributions by some of the leading experts in the field."—Professor Giuseppe Gaeta, Università di Milano

• Part A: Integrability, Lax Pairs and Symmetry. A1. Reciprocal transformations and their role in the integrability and classification of PDEs. A2. Contact Lax pairs and associated (3+1)-dimensional integrable dispersionless systems. A3. Lax Pairs for Edge-constrained Boussinesq Systems of Partial Difference Equations. A4. Lie point symmetries of delay ordinary differential equations. A5. The symmetry approach to integrability: recent advances. A6. Evolution of the concept of λ--symmetry and main applications. A7. Heir-equations for partial dfferential equations: a 25-year review. Part B: Algebraic and Geometric Methods. B1. Coupled nonlinear Schrodinger equations: spectra and instabilities of plane waves. B2. Rational solutions of Painleve systems. B3. Cluster algebras and discrete integrability. B4. A review of elliptic difference Painleve equations. B5. Linkage mechanisms governed by integrable deformations of discrete space curves. B6. The Cauchy problem of the Kadomtsev-Petviashvili hierarchy and infinite-dimensional groups. B7. Wronskian solutions of integrable systems. Part C: Applications. C1. Global gradient catastrophe in a shallow water model: evolution unfolding by stretched coordiates. C2. Vibrations of an elastic bar, isospectral deformations, and modified Camassa-Holm equations. C3. Exactly solvable (discrete) quantum mechanics and new orthogonal polynomials.