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Symmetries and Integrability of Difference Equations [Minkštas viršelis]

Edited by (University of Minnesota), Edited by , Edited by (Université de Montréal), Edited by (Universitą degli Studi Roma Tre)
  • Formatas: Paperback / softback, 360 pages, aukštis x plotis x storis: 228x151x18 mm, weight: 510 g, 40 Halftones, black and white
  • Serija: London Mathematical Society Lecture Note Series
  • Išleidimo metai: 23-Jun-2011
  • Leidėjas: Cambridge University Press
  • ISBN-10: 052113658X
  • ISBN-13: 9780521136587
Kitos knygos pagal šią temą:
Symmetries and Integrability of Difference EquationsDidesnis paveikslėlis
  • Formatas: Paperback / softback, 360 pages, aukštis x plotis x storis: 228x151x18 mm, weight: 510 g, 40 Halftones, black and white
  • Serija: London Mathematical Society Lecture Note Series
  • Išleidimo metai: 23-Jun-2011
  • Leidėjas: Cambridge University Press
  • ISBN-10: 052113658X
  • ISBN-13: 9780521136587
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9780521136587.html
Raktažodžiai:
"Difference equations are playing an increasingly important role in the natural sciences. Indeed many phenomena are inherently discrete and are naturally described by difference equations. Phenomena described by differential equations are therefore approximations of more basic discrete ones. Moreover, in their study it is very often necessary to resort to numerical methods. This always involves a discretization of the differential equations involved, thus replacing them by difference equations. This bookshows how Lie group and integrability techniques, originally developed for differential equations, have been adapted to the case of difference ones. Each of the eleven chapters is a self-contained treatment of a topic, containing introductory material aswell as the latest research results. The book will be welcomed by graduate students and researchers seeking an introduction to the field. As a survey of the current state of the art it will also serve as a valuable reference"--

Provided by publisher.

Recenzijos

'The book can certainly be recommended to everyone interested in having a comprehensive overview of the current state of the art in this fascinating and important research field.' Frans Cantrijn, Mathematical Reviews

Daugiau informacijos

A comprehensive introduction to and survey of the state of the art, suitable for graduate students and researchers alike.
List of figures
xi
List of contributors
xiv
Preface xvii
Introduction 1(6)
1 Lagrangian and Hamiltonian Formalism for Discrete Equations: Symmetries and First Integrals
7(43)
V. Dorodnitsyn
R. Kozlov
1.1 Introduction
7(2)
1.2 Invariance of Euler-Lagrange equations
9(2)
1.3 Lagrangian formalism for second-order difference equations
11(5)
1.4 Hamiltonian formalism for differential equations
16(5)
1.4.1 Canonical Hamiltonian equations
16(2)
1.4.2 The Legendre transformation
18(1)
1.4.3 Invariance of canonical Hamiltonian equations
18(3)
1.5 Discrete Hamiltonian formalism
21(10)
1.5.1 Discrete Legendre transform
21(2)
1.5.2 Variational formulation of the discrete Hamiltonian equations
23(1)
1.5.3 Symplecticity of the discrete Hamiltonian equations
24(3)
1.5.4 Invariance of the Hamiltonian action
27(1)
1.5.5 Discrete Hamiltonian identity and discrete Noether theorem
28(2)
1.5.6 Invariance of the discrete Hamiltonian equations
30(1)
1.6 Examples
31(15)
1.6.1 Nonlinear motion
31(5)
1.6.2 A nonlinear ODE
36(3)
1.6.3 Discrete harmonic oscillator
39(4)
1.6.4 Modified discrete harmonic oscillator (exact scheme)
43(3)
1.7 Conclusion
46(4)
2 Painleve Equations: Continuous, Discrete and Ultradiscrete
50(33)
B. Grammaticos
A. Ramani
2.1 Introduction
50(1)
2.2 A rough sketch of the top-down description of the Painleve equations
51(6)
2.3 A succinct presentation of the bottom-up description of the Painleve equations
57(7)
2.4 Properties of the, continuous and discrete, Painleve equations: a parallel presentation
64(10)
2.4.1 Degeneration cascade
64(1)
2.4.2 Lax pairs
65(2)
2.4.3 Miura and Backlund relations
67(2)
2.4.4 Particular solutions
69(3)
2.4.5 Contiguity relations
72(2)
2.5 The ultradiscrete Painleve equations
74(6)
2.5.1 Degeneration cascade
75(1)
2.5.2 Lax pairs
76(1)
2.5.3 Miura and Backlund relations
76(2)
2.5.4 Particular solutions
78(1)
2.5.5 Contiguity relations
78(2)
2.6 Conclusion
80(3)
3 Definitions and Predictions of Integrability for Difference Equations
83(32)
J. Hietarinta
3.1 Preliminaries
83(5)
3.1.1 Points of view on integrability
83(1)
3.1.2 Preliminaries on discreteness and discrete integrability
84(4)
3.2 Conserved quantities
88(3)
3.2.1 Constants of motion for continuous ODE
88(1)
3.2.2 The standard discrete case
89(1)
3.2.3 The Hirota-Kimura-Yahagi (HKY) generalization
90(1)
3.3 Singularity confinement and algebraic entropy
91(6)
3.3.1 Singularity analysis for difference equations
91(3)
3.3.2 Singularity confinement in projective space
94(2)
3.3.3 Singularity confinement is not sufficient
96(1)
3.4 Integrability in 2D
97(4)
3.4.1 Definitions and examples
97(1)
3.4.2 Quadrilateral lattices
98(2)
3.4.3 Continuum limit
100(1)
3.4.4 Conservation laws
100(1)
3.5 Singularity confinement in 2D
101(2)
3.6 Algebraic entropy for 2D lattices
103(3)
3.6.1 Default growth of degree and factorization
103(2)
3.6.2 Search based on factorization
105(1)
3.7 Consistency around a cube
106(3)
3.7.1 Definition
106(1)
3.7.2 Lax pair
107(1)
3.7.3 CAC as a search method
108(1)
3.8 Soliton solutions
109(3)
3.8.1 Background solutions
109(1)
3.8.2 1SS
110(1)
3.8.3 NSS
111(1)
3.9 Conclusions
112(3)
4 Orthogonal Polynomials, their Recursions, and Functional Equations
115(24)
M.E.H. Ismail
4.1 Introduction
115(1)
4.2 Orthogonal polynomials
116(3)
4.3 The spectral theorem
119(2)
4.4 The Freud nonlinear recursions
121(1)
4.5 Differential equations
122(3)
4.6 q-difference equations
125(3)
4.7 The inverse problem
128(2)
4.8 Applications
130(4)
4.9 The Askey-Wilson polynomials
134(5)
5 Discrete Painleve Equations and Orthogonal Polynomials
139(21)
A. Its
5.1 General setting
139(7)
5.1.1 Orthogonal polynomials
140(1)
5.1.2 Connections to integrable systems
140(2)
5.1.3 The Riemann-Hilbert representation of the orthogonal polynomials
142(1)
5.1.4 Discrete Painleve equations
143(3)
5.2 Examples
146(14)
5.2.1 Gaussian weight
146(1)
5.2.2 d-Painleve I
147(5)
5.2.3 d-Painleve XXXIV
152(8)
6 Generalized Lie Symmetries for Difference Equations
160(31)
D. Levi
R. I. Yamilov
6.1 Introduction
160(4)
6.1.1 Direct construction of generalized symmetries: an example
161(3)
6.2 Generalized symmetries from the integrability properties
164(13)
6.2.1 Toda Lattice
164(4)
6.2.2 The symmetry algebra for the Toda Lattice
168(2)
6.2.3 The continuous limit of the Toda symmetry algebras
170(2)
6.2.4 Backlund transformations for the Toda equation
172(1)
6.2.5 Backlund transformations vs. generalized symmetries
173(2)
6.2.6 Generalized symmetries for PΔE's
175(2)
6.3 Formal symmetries and integrable lattice equations
177(14)
6.3.1 Formal symmetries and further integrability conditions
182(3)
6.3.2 Why integrable equations on the lattice must be symmetric
185(3)
6.3.3 Example of classification problem
188(3)
7 Four Lectures on Discrete Systems
191(16)
S. P. Novikov
7.1 Introduction
191(1)
7.2 Discrete symmetries and completely integrable systems
191(2)
7.3 Discretization of linear operators
193(2)
7.4 Discrete GLn connections and triangle equation
195(3)
7.5 New discretization of complex analysis
198(9)
8 Lectures on Moving Frames
207(40)
P. J. Olver
8.1 Introduction
207(2)
8.2 Equivariant moving frames
209(2)
8.3 Moving frames on jet space and differential invariants
211(2)
8.4 Equivalence and signatures
213(2)
8.5 Joint invariants and joint differential invariants
215(3)
8.6 Invariant numerical approximations
218(6)
8.7 The invariant bicomplex
224(7)
8.8 Generating differential invariants
231(4)
8.9 Invariant variational problems
235(2)
8.10 Invariant curve flows
237(10)
9 Lattices of Compact Semisimple Lie Groups
247(12)
J. Patera
9.1 Introduction
247(1)
9.2 Motivating example
248(2)
9.3 Simple Lie groups and simple Lie algebras
250(4)
9.3.1 Simple roots
250(1)
9.3.2 Standard bases in Rn
251(1)
9.3.3 Reflections and affine reflections in Rn
252(1)
9.3.4 Weyl group and Affine Weyl group
252(2)
9.4 Lattice grids FM ⊂ F ⊂ Rn
254(2)
9.4.1 Examples of FM
255(1)
9.5 W-invariant functions orthogonal on FM
256(1)
9.6 Properties of elements of finite order
257(2)
10 Lectures on Discrete Differential Geometry
259(33)
Yu. B Suris
10.1 Basic notions
260(4)
10.2 Backlund transformations
264(2)
10.3 Q-nets
266(2)
10.4 Circular nets
268(3)
10.5 Q-nets in quadrics
271(2)
10.6 T-nets
273(3)
10.7 A-nets
276(5)
10.8 T-nets in quadrics
281(2)
10.9 K-nets
283(3)
10.10 Hirota equation for K-nets
286(6)
11 Symmetry Preserving Discretization of Differential Equations and Lie Point Symmetries of Differential-Difference Equations
292
P. Winternitz
11.1 Symmetry preserving discretization of ODEs
294(6)
11.1.1 Formulation of the problem
294(1)
11.1.2 Lie point symmetries of ordinary difference schemes
295(4)
11.1.3 The continuous limit
299(1)
11.2 Examples of symmetry preserving discretizations
300(6)
11.2.1 Equations invariant under SL1(2, R)
300(3)
11.2.2 Equations invariant under SL2(2, R)
303(1)
11.2.3 Equations invariant under the similitude group of the Euclidean plane
304(2)
11.3 Applications to numerical solutions of ODEs
306(8)
11.3.1 General procedure for testing the numerical schemes
306(1)
11.3.2 Numerical experiments for a third-order ODE invariant under SL1(2, R)
307(3)
11.3.3 Numerical experiments for ODEs invariant under SL2(2, R)
310(3)
11.3.4 Numerical experiments for third-order ODE invariant under Sim(2)
313(1)
11.4 Exact solutions of invariant difference schemes
314(12)
11.4.1 Lagrangian formulation for second-order ODEs
315(3)
11.4.2 Lagrangian formulation for second order difference equations
318(3)
11.4.3 Example: Second-order ODE with three-dimensional solvable symmetry algebra
321(5)
11.5 Lie point symmetries of differential-difference equations
326(8)
11.5.1 Formulation of the problem
326(1)
11.5.2 The evolutionary formalism and commuting flows for differential equations
327(1)
11.5.3 The evolutionary formalism and commuting flows for differential-difference equations
328(2)
11.5.4 General algorithm for calculating Lie point symmetries of a differential-difference equation
330(1)
11.5.5 Theorems simplifying the calculation of symmetries of DΔE
330(1)
11.5.6 Volterra type equations and their generalizations
331(2)
11.5.7 Toda type equations
333(1)
11.5.8 Toda field theory type equations
333(1)
11.6 Examples of symmetries of DΔE
334
11.6.1 The YdKN equation
335(1)
11.6.2 The Toda lattice
336(1)
11.6.3 The two-dimensional Toda lattice equation
336
Decio Levi is a researcher in the Faculty of Engineering at the Universitą degli Studi Roma Tre. Peter Olver is a Professor and currently Head of the School of Mathematics at the University of Minnesota. Zora Thomova is an Associate Professor of Mathematics at the State University of New York Institute of Technology. Pavel Winternitz is a Professor in the Department of Mathematics and Statistics at the Université de Montréal.