1st Edition

# Nonlinear Systems and Their Remarkable Mathematical Structures, Volumes 1, 2, and 3

1648 Pages 62 Color & 77 B/W Illustrations
by Chapman & Hall

This set of three volumes 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. These volumes should be suitable for 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 all the volumes 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.

Features

• Clearly illustrates the mathematical theories of nonlinear systems and their progress to both the non-expert and active researchers in this area.
• Suitable for graduate students in mathematics, applied mathematics and some of the engineering sciences.
• Written in a careful pedagogical manner by those experts who have been involved in the research themselves, with each contribution being reasonably self-contained.

Table of Content - 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. A4. Elementary introduction to discrete soliton equations. A5. New results on integrability of the Kahan-Hirota-Kimura discretizations. 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. 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 Ito stochastic differential equations and their applications. C5. Statistical symmetries of turbulence. 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.

Table of Content - Volume 2

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.

Table of Content - Volume 3

Part A: Integrability and Symmetries. A1. The BKP hierarchy and the modified BKP hierarchy. A2. Elementary introduction to the direct linearisation of integrable systems. A3. Discrete Boussinesq-type equations. A4. The study of integrable hierarchies in terms of Liouville correspondences. A5. Darboux transformations for supersymmetric integrable systems: A brief review. A6. Nonlocal symmetries of nonlinear integrable systems. A7. High-order soliton matrix for an extended nonlinear Schrödinger equation. A8. Darboux transformation for integrable systems with symmetries. A9. Frobenius manifolds and Orbit spaces of reflection groups and their extensions. Part B: Algebraic, Analytic and Geometric Methods. B1. On finite Toda type lattices and multipeakons of the Camassa-Holm type equations. B2. Long-time asymptotics for the generalized coupled derivative nonlinear Schrödinger equation. B3. Bilinearization of nonlinear integrable evolution equations: Recursion operator approach. B4. Rogue wave patterns and modulational instability in nonlinear Schrödinger hierarchy. B5. Algebro-geometric solutions to the modified Blaszak-Marciniak lattice hierarchy. B6. Long-time asymptotic behavior of the modified Schrödinger equation via θ-steepest descent method. B7. Two hierarchies of multiple solitons and soliton molecules of (2+1)-dimensional Sawada-Kotera type equation. B8. Dressing the boundary: exact solutions of soliton equations on the half-line. B9. From integrable spatial discrete hierarchy to integrable nonlinear PDE hierarchy.

### Biography

Norbert Euler is currently a visiting professor at the International Center of Sciences A.C. (Cuernavaca, Mexico). He has been teaching a wide variety of mathematics courses at both the undergraduate and postgraduate level at several universities worldwide for more than 25 years. He is an active researcher and has to date published over 80 peer reviewed research articles in the subject of nonlinear systems and is a co-author of several books. He is also involved in editorial work for some international journals.

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.

Da-Jun Zhang is currently a full professor at Shanghai University in China. His research focuses on integrability of discrete and continuous nonlinear systems, and particularly, discrete integrable systems. He has published over 120 peer reviewed research articles in the subject of integrable systems. He has served as scientific committee member for some international conferences. He is also involved in editorial work for some international journals