Nonlinear Systems and Their Remarkable Mathematical Structures Volume 1 1st Edition by Norbert Euler ISBN 9781138601000 1138601004

Part A: Nonlinear Integrable Systems

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

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

5. Extensions

A2. Integrable nonlinear PDEs on the half-line

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

1. Introduction

2. Singularity confinement

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

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

1. Introduction

2. General properties of the Kahan-Hirota-Kimura discretization

3. Novel observations and results

4. The general Clebsch flow

5. The first Clebsch flow

6. The Kirchhoff case

7. Lagrange top

8. Concluding remarks

Part B: Solution Methods and Solution Structures

B1. Solvable dynamical systems and isospectral matrices defined in terms of the zeros of orthogonal or otherwise special polynomials

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

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. Pfeiffer–Sato solutions of Buhl’s problem and a Lagrange–d’Alembert principle for heavenly equations

1. Introduction

2. Lax–Sato compatible systems of vector field 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 superflows: Their Lie-algebraic structure

9. Integrability and the Lagrange–d’Alembert principle

B4. Superposition formulae for nonlinear integrable equations in bilinear form

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

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

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 = 1

5. Final comments

Part C: Symmetry Methods for Nonlinear Systems

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

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

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

1. Notation and definitions

2. Symmetries

3. Recursions

4. Nonlocalities

C4. Symmetries of Itô stochastic differential equations and their applications

1. Introduction

2. Illustrating example

3. Itô SDEs and Lie point symmetries

4. Properties of symmetries of Itô SDEs

5. Symmetry applications

C5. Statistical symmetries of turbulence

1. Foreword

2. Stochastic behavior and symmetries of differential equations – an introduction

3. Statistics of the Navier-Stokes equations and its symmetries

4. Summary and outlook

Part D: Nonlinear Systems in Applications

D1. Integral transforms and ordinary differential equations of infinite order

1. Introduction

2. Differential operators of infinite 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 field equations

8. Future work

D2. On the rôle of nonlinearity in geostrophic ocean flows on a sphere

1. Introduction

2. Preliminaries

3. Governing equations

4. Geostrophy and the f- and β-plane approximations

5. Geostrophy in spherical coordinates

6. Discussion

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

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

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-Painlevé systems

Subject Index