دانلود کتاب Nonlinear Systems and Their Remarkable Mathematical Structures: Volume 1

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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 connement
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 dened 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 innite order in mathematics and physics
3. Mathematical theory for nonlocal equations
4. The operator f(@t) : Lp(R+)