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E-raamat: Nonlinear Systems and Their Remarkable Mathematical Structures: Volume 1

Edited by Norbert Euler (Lulea University of Technology, Lulea)

Formaat: 598 pages
Ilmumisaeg: 19-Nov-2018
Kirjastus: CRC Press
Keel: eng
ISBN-13: 9780429893810

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Formaat: 598 pages
Ilmumisaeg: 19-Nov-2018
Kirjastus: CRC Press
Keel: eng
ISBN-13: 9780429893810

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Nonlinear Systems and Their Remarkable Mathematical Structures, Volume 1 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.

Features:

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

Arvustused

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

Preface

xiii

The Authors

Part A: Nonlinear Integrable Systems

A1 Systems of nonlinearly-coupled differential equations solvable by algebraic operations

(14)

F. Calogero

1 Introduction

(1)

2 The main idea and some key identities

(4)

3 Two examples of systems of nonlinearly-coupled ODEs solvable by algebraic operations

(4)

4 A differential algorithm to evaluate all the zeros of a generic polynomial of arbitrary degree

(2)

5 Extensions

(4)

A2 Integrable nonlinear PDEs on the half-line

(29)

A.S. Fokas

B. Pelloni

1 Introduction

(5)

2 Transforms and Riemann-Hilbert problems

(4)

3 The structure of integrable PDEs: Lax pair formulation

(2)

4 An integral transform for nonlinear boundary value problems

(11)

5 Further considerations

(7)

A3 Detecting discrete integrability: the singularity approach

(30)

B. Grammaticos

A. Ramani

R. Willox

T. Mase

1 Introduction

(2)

2 Singularity confinement

(7)

3 The full-deautonomisation approach

(4)

4 Halburd's exact calculation of the degree growth

(7)

5 Singularities and spaces of initial conditions

(10)

A4 Elementary introduction to discrete soliton equations

(20)

J. Hietarinta

1 Introduction

(1)

2 Basic set-up for lattice equations

(3)

3 Symmetries and hierarchies

(3)

4 Lax pairs

(2)

5 Continuum limits

(3)

6 Discretizing a continuous equation

(6)

7 Integrability test

(1)

8 Summary

(2)

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

(28)

Yu B. Suris

M. Petrera

1 Introduction

(2)

2 General properties of the Kahan-Hirota-Kimura discretization

(1)

3 Novel observations and results

(3)

4 The general Clebsch flow

(5)

5 The first Clebsch flow

104

(9)

6 The Kirchhoff case

113

(3)

7 Lagrange top

116

(1)

8 Concluding remarks

117

(5)

Part B. Solution Methods and Solution Structures

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

122

(37)

O. Bihun

1 Introduction

122

(5)

2 Zeros of generalized hypergeometric polynomial with two parameters and zeros of Jacobi polynomials

127

(6)

3 Zeros of generalized hypergeometric polynomials

133

(3)

4 Zeros of generalized basic hypergeometric polynomials

136

(5)

5 Zeros of Wilson and Racah polynomials

141

(6)

6 Zeros of Askey-Wilson and q-Racah polynomials

147

(7)

7 Discussion and Outlook

154

(5)

B2 Singularity methods for meromorphic solutions of differential equations

159

(28)

R. Conte

T.W. Ng

C.F. Wu

1 Introduction

159

(4)

2 A simple pedagogical example

163

(4)

3 Lessons from this pedagogical example

167

(2)

4 Another characterization of elliptic solutions: the subequation method

169

(3)

5 An alternative to the Hermite decomposition

172

(1)

6 The important case of amplitude equations

173

(6)

7 Nondegenerate elliptic solutions

179

(1)

8 Degenerate elliptic solutions

180

(2)

9 Current challenges and open problems

182

(5)

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

187

(46)

O.E. Hentosh

Ya A. Prykarpatsky

D. Blackmore

A. Prykarpatski

1 Introduction

187

(3)

2 Lax-Sato compatible systems of vector field equations

190

(5)

3 Heavenly equations: Lie-algebraic integrability scheme

195

(4)

4 Integrable heavenly dispersionless equations: Examples

199

(3)

5 Lie-algebraic structures and heavenly dispersionless systems

202

(5)

6 Linearization covering method and its applications

207

(8)

7 Contact geometry linearization covering scheme

215

(2)

8 Integrable heavenly superflows: Their Lie-algebraic structure

217

(5)

9 Integrability and the Lagrange-d'Alembert principle

222

(11)

B4 Superposition formulae for nonlinear integrable equations in bilinear form

233

(24)

X.B. Hu

1 Introduction

233

(2)

2 Bianchi theorem of permutability and superposition formula of the KdV equation

235

(2)

3 Superposition formulae for a variety of soliton equations with examples

237

(10)

4 Superposition formulae for rational solutions

247

(5)

5 Superposition formulae for some other particular solutions

252

(5)

B5 Matrix solutions for equations of the AKNS system

257

(38)

C. Schiebold

1 Introduction

257

(2)

2 An operator approach to integrable systems

259

(4)

3 The nc AKNS system

263

(2)

4 Solution formulas for the AKNS system

265

(3)

5 Projection techniques revisited

268

(1)

6 Matrix- and vector-AKNS systems

269

(3)

7 Reduction

272

(1)

8 The finite-dimensional case

273

(5)

9 Solitons, strongly bound solitons (breathers), degeneracies

278

(4)

10 Multiple pole solutions

282

(5)

11 Solitons of matrix- and vector-equations

287

(8)

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

295

(22)

C. Valls

1 Introduction and statement of the main results

295

(4)

2 Proof of Theorem 2 and some preliminary results

299

(2)

3 Proof of Theorem 3 with n = 1

301

(6)

4 Proof of Theorem 3 with n = 2

307

(6)

5 Final comments

313

(4)

Part C: Symmetry Methods for Nonlinear Systems

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

317

(35)

M. Euler

N. Euler

1 Introduction: symmetry-integrable equations and multipotentialisations

317

(10)

2 The multipotentialisation of the Kupershmidt equation

327

(6)

3 Invariance of the Kupershmidt equation and its chain of potentialisations

333

(5)

4 The hierarchies

338

(4)

5 Concluding remarks

342

(1)

Appendix A: A list of recursion operators

343

(5)

Appendix B: An equation that does not potentialise

348

(4)

C2 Geometry of normal forms for dynamical systems

352

(38)

G. Gaeta

1 Introduction

352

(2)

2 Normal forms

354

(2)

3 Normal forms and symmetry

356

(2)

4 Michel theory

358

(2)

5 Unfolding of normal forms

360

(4)

6 Normal forms in the presence of symmetry

364

(1)

7 Normal forms and classical Lie groups

365

(3)

8 Finite normal forms

368

(1)

9 Gradient property

369

(1)

10 Spontaneous linearization

370

(1)

11 Discussion and conclusions

371

(2)

Appendix A: The normal forms construction

373

(3)

Appendix B: Examples of unfolding

376

(3)

Appendix C: Hopf and Hamiltonian Hopf bifurcations

379

(2)

Appendix D: Symmetry and convergence for normal forms

381

(9)

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

390

(18)

A.V. Kiselev

A.O. Krutov

T. Wolf

1 Notation and definitions

391

(2)

2 Symmetries

393

(3)

3 Recursions

396

(3)

4 Nonlocalities

399

(9)

C4 Symmetries of Ito stochastic differential equations and their applications

408

(29)

R. Kozlov

1 Introduction

408

(1)

2 Illustrating example

409

(1)

3 Ito SDEs and Lie point symmetries

410

(3)

4 Properties of symmetries of Ito SDEs

413

(7)

5 Symmetry applications

420

(17)

C5 Statistical symmetries of turbulence

437

(31)

M. Oberlack

M. Waclawczyk

V. Grebenev

1 Foreword

437

(1)

2 Stochastic behavior and symmetries of differential equations - an introduction

438

(6)

3 Statistics of the Navier-Stokes equations and its symmetries

444

(20)

4 Summary and outlook

464

(4)

Part D: Nonlinear Systems in Applications

D1 Integral transforms and ordinary differential equations of infinite order

468

(32)

A. Chavez

H. Prado

E.G. Reyes

1 Introduction

468

(2)

2 Differential operators of infinite order in mathematics and physics

470

(3)

3 Mathematical theory for nonlocal equations

473

(5)

4 The operator f (partialdifferentialt) : LP(R+) right arrow Hq(C+)

478

(5)

5 The initial value problem

483

(5)

6 From the Laplace to the Borel transform

488

(4)

7 Linear zeta-nonlocal field equations

492

(3)

8 Future work

495

(5)

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

500

(20)

A. Constantin

R.S. Johnson

1 Introduction

500

(1)

2 Preliminaries

501

(1)

3 Governing equations

501

(2)

4 Geostrophy and the f- and β-plane approximations

503

(7)

5 Geostrophy in spherical coordinates

510

(6)

6 Discussion

516

(4)

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

520

(21)

A.V. Osipov

G. Soderbacka

1 Introduction

520

(1)

2 Lotka-Volterra equations

521

(2)

3 Rosenzweig-McArthur equations

523

(1)

4 Mathematical tools

524

(4)

5 Systems with more predators

528

(9)

6 Modified standard system

537

(4)

D4 Ermakov-type systems in nonlinear physics and continuum mechanics

541

(36)

C. Rogers

W.K. Schief

1 Overview

541

(3)

2 A rotating shallow water system. Ermakov-Ray-Reid reduction

544

(4)

3 Hamiltonian Ermakov-Ray-Reid reduction in magneto-gasdynamics. The pulsrodon

548

(5)

4 Hamiltonian Ermakov-Ray-Reid systems. Parametrisation and integration

553

(2)

5 Multi-component Ermakov systems. Genesis in N-layer hydrodynamics

555

(4)

6 Multi-component Ermakov and many-body system connections

559

(3)

7 Multi-component Ermakov-Painleve systems

562

(15)

Subject Index

577

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.

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