Nonlinear Systems and Their Remarkable Mathematical Structures: Volume 1

Description

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 Cover Half Title Title Copyrights Contents Preface The Authors Part A A1. Systems of nonlinearly-coupled di erential equations solvable by algebraic operations F Calogero 1. Introduction 2. The main idea and some key identities 3. Two examples of systems of nonlinearly-coupled ODEs solvable by algebraic operations 4. A dierential algorithm to evaluate all the zeros of a generic polynomial of arbitrary degree 5. Extensions A2. Integrable nonlinear PDEs on the half-line A S Fokas and B Pelloni 1. Introduction 2. Transforms and Riemann-Hilbert problems 3. The structure of integrable PDEs: Lax pair formulation 4. An integral transform for nonlinear boundary value problems 5. Further considerations A3. Detecting discrete integrability: the singularity approach B Grammaticos, A Ramani, R Willox and T Mase 1. Introduction 2. Singularity connement 3. The full-deautonomisation approach 4. Halburd's exact calculation of the degree growth 5. Singularities and spaces of initial conditions A4. Elementary introduction to discrete soliton equations J Hietarinta 1. Introduction 2. Basic set-up for lattice equations 3. Symmetries and hierarchies 4. Lax pairs 5. Continuum limits 6. Discretizing a continuous equation 7. Integrability test 8. Summary A5. New results on integrability of the Kahan-Hirota-Kimura discretizations Yu B Suris and M Petrera 1. Introduction 2. General properties of the Kahan-Hirota-Kimura discretization 3. Novel observations and results 4. The general Clebsch ow 5. The rst Clebsch ow 6. The Kirchho case 7. Lagrange top 8. Concluding remarks Part B B1. Dynamical systems satisfied by special polynomials and related isospectral matrices dened in terms of their zeros O Bihun 1. Introduction 2. Zeros of generalized hypergeometric polynomial with two parameters and zeros of Jacobi polynomials 3. Zeros of generalized hypergeometric polynomials 4. Zeros of generalized basic hypergeometric polynomials 5. Zeros of Wilson and Racah polynomials 6. Zeros of Askey-Wilson and q-Racah polynomials 7. Discussion and Outlook B2. Singularity methods for meromorphic solutions of differential equations R Conte, T W Ng and C F Wu 1. Introduction 2. A simple pedagogical example 3. Lessons from this pedagogical example 4. Another characterization of elliptic solutions: the subequation method 5. An alternative to the Hermite decomposition 6. The important case of amplitude equations 7. Nondegenerate elliptic solutions 8. Degenerate elliptic solutions 9. Current challenges and open problems B3. Pfei er-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 1. Introduction 2. Lax{Sato compatible systems of vector eld equations 3. Heavenly equations: Lie-algebraic integrability scheme 4. Integrable heavenly dispersionless equations: Examples 5. Lie-algebraic structures and heavenly dispersionless systems 6. Linearization covering method and its applications 7. Contact geometry linearization covering scheme 8. Integrable heavenly super ows: Their Lie-algebraic structure 9. Integrability and the Lagrange{d'Alembert principle B4. Superposition formulae for nonlinear integrable equations in bilinear form X B Hu 1. Introduction 2. Bianchi theorem of permutability and superposition formula of the KdV equation 3. Superposition formulae for a variety of soliton equations with examples 4. Superposition formulae for rational solutions 5. Superposition formulae for some other particular solutions B5. Matrix solutions for equations of the AKNS system C Schiebold 1. Introduction 2. An operator approach to integrable systems 3. The nc AKNS system 4. Solution formulas for the AKNS system 5. Projection techniques revisited 6. Matrix- and vector-AKNS systems 7. Reduction 8. The finite-dimensional case 9. Solitons, strongly bound solitons (breathers), degeneracies 10. Multiple pole solutions 11. Solitons of matrix- and vector-equations B6. Algebraic traveling waves for the generalized KdV-Burgers equation and the Kuramoto-Sivashinsky equation C Valls 1. Introduction and statement of the main results 2. Proof of Theorem 2 and some preliminary results 3. Proof of Theorem 3 with n = 1 4. Proof of Theorem 3 with n = 2 5. Final comments Part C C1. Nonlocal invariance of the multipotentialisations of the Kupershmidt equation and its higher-order hierarchies M Euler and N Euler 1. Introduction: symmetry-integrable equations and multipotentialisations 2. The multipotentialisation of the Kupershmidt equation 3. Invariance of the Kupershmidt equation and its chain of potentialisations 4. The hierarchies 5. Concluding remarks Appendix A: A list of recursion operators Appendix B: An equation that does not potentialise C2. Geometry of normal forms for dynamical systems G Gaeta 1. Introduction 2. Normal forms 3. Normal forms and symmetry 4. Michel theory 5. Unfolding of normal forms 6. Normal forms in the presence of symmetry 7. Normal forms and classical Lie groups 8. Finite normal forms 9. Gradient property 10. Spontaneous linearization 11. Discussion and conclusions Appendix A: The normal forms construction Appendix B: Examples of unfolding Appendix C: Hopf and Hamiltonian Hopf bifurcations Appendix D: Symmetry and convergence for normal forms C3. Computing symmetries and recursion operators of evolutionary super-systems using the SsTools environment A V Kiselev, A O Krutov and T Wolf 1. Notation and definitions 2. Symmetries 3. Recursions 4. Nonlocalities C4. Symmetries of It^o stochastic di erential equations and their applications R Kozlov 1. Introduction 2 Illustrating example 3. It^o SDEs and Lie point symmetries 4. Properties of symmetries of Ito ^SDEs 5. Symmetry applications C5. Statistical symmetries of turbulence M Oberlack, M Wac lawczyk and V Grebenev 1. Foreword 2. Stochastic behavior and symmetries of dierential equations - an introduction 3. Statistics of the Navier-Stokes equations and its symmetries 4. Summary and outlook Part D D1. Integral transforms and ordinary di erential equations of infinite order A Chavez, H Prado and E G Reyes 1. Introduction 2. Dierential operators of innite order in mathematics and physics 3. Mathematical theory for nonlocal equations 4. The operator f(@t) : Lp(R+) 􀀀! Hq(C+) 5. The initial value problem 6. From the Laplace to the Borel transform 7. Linear zeta-nonlocal eld equations 8. Future work D2. The role of nonlinearity in geostrophic ocean ows on a sphere A Constantin and R S Johnson 1. Introduction 2. Preliminaries 3. Governing equations 4. Geostrophy and the f- and -plane approximations 5. Geostrophy in spherical coordinates 6. Discussion D3. Review of results on a system of type many predators - one prey A V Osipov and G SŁoderbacka 1. Introduction 2. Lotka-Volterra equations 3. Rosenzweig-McArthur equations 4. Mathematical tools 5. Systems with more predators 6. Modified standard system D4. Ermakov-type systems in nonlinear physics and continuum mechanics C Rogers and W K Schief 1. Overview 2. A rotating shallow water system. Ermakov-Ray-Reid reduction 3. Hamiltonian Ermakov-Ray-Reid reduction in magneto-gasdynamics. The pulsrodon 4. Hamiltonian Ermakov-Ray-Reid systems. Parametrisation and integration 5. Multi-component Ermakov systems. Genesis in N-layer hydrodynamics 6. Multi-component Ermakov and many-body system connections 7. Multi-component Ermakov-Painleve systems Subject Index