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E-grāmata: Discrete Systems and Integrability

(University of Turku, Finland), (University of Sydney), (University of Leeds)
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This first introductory text to discrete integrable systems introduces key notions of integrability from the vantage point of discrete systems, also making connections with the continuous theory where relevant. While treating the material at an elementary level, the book also highlights many recent developments. Topics include: Darboux and Bäcklund transformations; difference equations and special functions; multidimensional consistency of integrable lattice equations; associated linear problems (Lax pairs); connections with Padé approximants and convergence algorithms; singularities and geometry; Hirota's bilinear formalism for lattices; intriguing properties of discrete Painlevé equations; and the novel theory of Lagrangian multiforms. The book builds the material in an organic way, emphasizing interconnections between the various approaches, while the exposition is mostly done through explicit computations on key examples. Written by respected experts in the field, the numerous exercises and the thorough list of references will benefit upper-level undergraduate, and beginning graduate students as well as researchers from other disciplines.

Papildus informācija

A first introduction to the theory of discrete integrable systems at a level suitable for students and non-experts.
Preface xi
1 Introduction to difference equations
1(30)
1.1 A first look at discrete equations
1(15)
1.2 The Riccati equation
16(5)
1.3 Partial difference equations
21(6)
1.4 Notes
27(4)
Exercises
28(3)
2 Discrete equations from transformations of continuous equations
31(36)
2.1 Special functions and linear equations
32(9)
2.2 Addition formulae
41(3)
2.3 The Painleve equations
44(6)
2.4 Backlund transformations for nonlinear PDEs
50(3)
2.5 Infinite sequence of conservation laws and KdV hierarchy
53(8)
2.6 Notes
61(6)
Exercises
62(5)
3 Integrability of PΔEs
67(52)
3.1 Quadrilateral PΔEs
68(3)
3.2 Consistency-around-the-cube as integrability
71(4)
3.3 Lax pairs and Backlund transformation from CAC
75(7)
3.4 Yang--Baxter maps
82(4)
3.5 Classification of quadrilateral PΔEs
86(4)
3.6 Different equations on different faces of the consistency cube
90(4)
3.7 CAC for multi-component equations
94(9)
3.8 Lattice KdV, SKdV and mKdV equations
103(7)
3.9 Higher-dimensional equations: the KP class
110(4)
3.10 Notes
114(5)
Exercises
116(3)
4 Interlude: Lattice equations and numerical algorithms
119(17)
4.1 Pade approximants
119(7)
4.2 Convergence acceleration algorithm
126(2)
4.3 Rutishauser's QD algorithm
128(5)
4.4 Notes
133(3)
Exercises
134(2)
5 Continuum limits of lattice PΔE
136(23)
5.1 How to take a continuum limit
136(1)
5.2 Plane-wave factors and linearization
137(2)
5.3 The semi-continuous limits
139(5)
5.4 Semi-discrete Lax pairs
144(3)
5.5 Full continuum limit
147(4)
5.6 All at once, or the double continuum limit
151(1)
5.7 Continuum limits of the 9-point BSQ
152(2)
5.8 Notes
154(5)
Exercises
155(4)
6 One-dimensional lattices and maps
159(38)
6.1 Integrability of maps
159(9)
6.2 The Kahan--Hirota--Kimura discretization
168(1)
6.3 The QRT maps
169(8)
6.4 Periodic reductions
177(8)
6.5 Lax pair for the periodic reductions and construction of invariants
185(4)
6.6 Pole reduction of the semi-discrete KP equation
189(4)
6.7 Notes
193(4)
Exercises
194(3)
7 Identifying integrable difference equations
197(26)
7.1 Singularity analysis of differential and difference equations
197(10)
7.2 Algebraic entropy
207(6)
7.3 Singularities from a geometric point of view
213(6)
7.4 Notes
219(4)
Exercises
221(2)
8 Hirota's bilinear method
223(27)
8.1 Introduction
223(3)
8.2 Soliton solutions
226(3)
8.3 Hirota's and Miwa's equations
229(4)
8.4 Reductions of the Hirota-Miwa equation
233(5)
8.5 Bilinearization of a lattice equation
238(3)
8.6 Solutions in matrix form
241(4)
8.7 Notes
245(5)
Exercises
246(4)
9 Multi-soliton solutions and the Cauchy matrix scheme
250(30)
9.1 Cauchy matrix structure for KdV-type equations
250(5)
9.2 Closed-form lattice equations
255(2)
9.3 Derivation of Lax pairs
257(4)
9.4 Bilinear form from soliton solutions
261(5)
9.5 The NQC and Q3 equations
266(2)
9.6 Proof of the Q3 N-soliton solution
268(4)
9.7 Higher-dimensional soliton systems: the KP class
272(5)
9.8 Notes
277(3)
Exercises
277(3)
10 Similarity reductions of integrable PΔEs
280(24)
10.1 Introduction to dimensional reductions
280(4)
10.2 Compatibility of lattice constraint with quad equations
284(1)
10.3 The linear case
285(4)
10.4 Similarity constraints for the lattice KdV family
289(11)
10.5 Notes
300(4)
Exercises
301(3)
11 Discrete Painleve equations
304(30)
11.1 Early discoveries of discrete Painleve equations
305(2)
11.2 Discrete Painleve equations from Sakai's classification
307(3)
11.3 Coalescences and degeneracies of the discrete Painleve equations
310(1)
11.4 Backlund and other transformations of discrete Painleve equations
311(4)
11.5 Affine Weyl groups
315(7)
11.6 Linear problems
322(4)
11.7 Linearization of discrete Painleve equations
326(2)
11.8 Sakai's elliptic discrete Painleve equation
328(1)
11.9 Notes
329(5)
Exercises
331(3)
12 Lagrangian multiform theory
334(29)
12.1 Conventional Lagrange theory and its discrete analogue
335(8)
12.2 Lagrangian 2-form structure
343(9)
12.3 Lagrangian 1-form structure
352(7)
12.4 Notes
359(4)
Exercises
360(3)
Appendix A Elementary difference calculus and difference equations 363(21)
Appendix B Theta functions and elliptic functions 384(20)
Appendix C The continuous Painleve equations and the Garnier system 404(3)
Appendix D Some determinantal identities 407(4)
References 411(29)
Index 440
J. Hietarinta is Professor Emeritus of Theoretical Physics at the University of Turku, Finland. His work has focused on the search for integrable systems of various forms, including Hamiltonian mechanics, Hirota bilinear form, Yang-Baxter and tetrahedron equations, as well as lattice equations. He was instrumental in the setting up of the nlin.SI category in arxiv.org and created the web pages for the SIDE (Symmetries and Integrability of Difference Equations) conference series (http://side-conference.net). N. Joshi is Professor of Applied Mathematics at the University of Sydney. She is best known for her work on the Painlevé equations and works at the leading edge of international efforts to analyse discrete and continuous integrable systems in the geometric setting of their initial-value spaces, constructed by resolving singularities in complex projective space. She was elected as a Fellow of the Australian Academy of Science in 2008, holds a Georgina Sweet Australian Laureate Fellowship and was awarded the special Hardy Fellowship of the London Mathematical Society in 2015. F. W. Nijhoff is Professor of Mathematical Physics in the School of Mathematics at the University of Leeds. His research focuses on nonlinear difference and differential equations, symmetries and integrability of discrete systems, variational calculus, quantum integrable systems and linear and nonlinear special functions. He was the principal organizer of the 2009 six-month programme on Discrete Integrable Systems at the Isaac Newton Institute, and was awarded a Royal Society Leverhulme Trust Senior Research Fellow in 2011.