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Continuous Symmetries and Integrability of Discrete Equations [Kietas viršelis]

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  • Formatas: Hardback, 496 pages, aukštis x plotis: 254x178 mm, weight: 479 g
  • Serija: CRM Monograph Series
  • Išleidimo metai: 15-Apr-2023
  • Leidėjas: American Mathematical Society
  • ISBN-10: 0821843540
  • ISBN-13: 9780821843543
Kitos knygos pagal šią temą:
Continuous Symmetries and Integrability of Discrete EquationsDidesnis paveikslėlis
  • Formatas: Hardback, 496 pages, aukštis x plotis: 254x178 mm, weight: 479 g
  • Serija: CRM Monograph Series
  • Išleidimo metai: 15-Apr-2023
  • Leidėjas: American Mathematical Society
  • ISBN-10: 0821843540
  • ISBN-13: 9780821843543
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9780821843543.html
Raktažodžiai:
This book on integrable systems and symmetries presents new results on applications of symmetries and integrability techniques to the case of equations defined on the lattice. This relatively new field has many applications, for example, in describing the evolution of crystals and molecular systems defined on lattices, and in finding numerical approximations for differential equations preserving their symmetries.

The book contains three chapters and five appendices. The first chapter is an introduction to the general ideas about symmetries, lattices, differential difference and partial difference equations and Lie point symmetries defined on them. Chapter 2 deals with integrable and linearizable systems in two dimensions. The authors start from the prototype of integrable and linearizable partial differential equations, the Korteweg de Vries and the Burgers equations. Then they consider the best known integrable differential difference and partial difference equations. Chapter 3 considers generalized symmetries and conserved densities as integrability criteria. The appendices provide details which may help the readers' understanding of the subjects presented in Chapters 2 and 3.

This book is written for PhD students and early researchers, both in theoretical physics and in applied mathematics, who are interested in the study of symmetries and integrability of difference equations.
Foreword xi
List of Figures
xiii
List of Tables
xv
Preface xvii
Acknowledgment xxi
Chapter 1 Introduction
1(50)
1 Lie point symmetries of differential equations, their extensions and applications
2(8)
2 What is a lattice
10(10)
2.1 1-dimensional lattices
10(1)
2.2 2-dimensional lattices
10(3)
2.3 Differential and difference operators on the lattice
13(1)
2.4 Grids and lattices in the description of difference equations
14(1)
2.4.1 Cartesian lattices
14(1)
2.4.2 Galilei invariant lattice
15(1)
2.4.3 Exponential lattice
15(1)
2.4.4 Polar coordinate systems
16(2)
2.5 Clairaut-Schwarz-Young theorem on the lattices and its consequences
18(1)
2.5.1 Commutativity and non commutativity of difference operators
18(2)
3 What is a difference equation
20(3)
3.1 Examples
22(1)
4 How do we find symmetries for difference equations
23(24)
4.1 Examples
26(1)
4.1.1 Lie point symmetries of the discrete time Toda lattice
26(2)
4.1.2 Lie point symmetries of DAEs
28(2)
4.1.3 Lie point symmetries of the Toda lattice
30(2)
4.1.4 Classification of DAEs
32(2)
4.1.5 Lie point symmetries of the two dimensional Toda equation
34(1)
4.2 Lie point symmetries preserving discretization of ODEs
35(3)
4.3 Group classification and solution of OAEs
38(1)
4.3.1 Symmetries of second order ODEs
38(2)
4.3.2 Symmetries of the three-point difference schemes
40(4)
4.3.3 Lagrangian formalism and solutions of three-point OAS
44(3)
5 What we leave out on symmetries in this book
47(1)
6 Outline of the book
48(3)
Chapter 2 Integrability and symmetries of nonlinear differential and difference equations in two independent variables
51(174)
1 Introduction
51(1)
2 Integrability of PDEs
52(34)
2.1 Introduction
52(1)
2.2 All you ever wanted to know about the integrability of the KdV equation and its hierarchy
53(4)
2.2.1 The KdV hierarchy: recursion operator
57(4)
2.2.2 The Backlund transformations, Darboux operators and Bianchi identity for the KdV hierarchy
61(3)
2.2.3 The conservations laws for the KdV equation
64(1)
2.2.4 The symmetries of the KdV hierarchy
65(1)
2.2.5 Lie algebra of the symmetries
66(2)
2.2.6 Relation between Backlund transformations and isospectral symmetries
68(2)
2.2.7 Symmetry reductions of the KdV equation
70(1)
2.3 The cylindrical KdV, its hierarchy and Darboux and Backlund transformations
71(3)
2.4 Integrable PDEs as infinite-dimensional superintegrable systems
74(3)
2.5 Integrability of the Burgers equation, the prototype of linearizable PDEs
77(2)
2.5.1 Backlund transformation and Bianchi identity for the Burgers hierarchy of equations
79(2)
2.5.2 Symmetries of the Burgers equation
81(1)
2.5.3 Symmetry reduction by Lie point symmetries
81(1)
2.6 General ideas on linearization
82(1)
2.6.1 Linearization of PDEs through symmetries
83(3)
3 Integrability of DAEs
86(43)
3.1 Introduction
86(1)
3.2 The Toda lattice, the Toda system, the Toda hierarchy and their symmetries
87(5)
3.2.1 Symmetries for the Toda hierarchy
92(1)
3.2.2 The Lie algebra of the symmetries for the Toda system and Toda lattice
93(3)
3.2.3 Contraction of the symmetry algebras in the continuous limit
96(1)
3.2.4 Backlund transformations and Bianchi identities for the Toda system and Toda lattice
97(3)
3.2.5 Relation between Backlund transformations and isospectral symmetries
100(2)
3.2.6 Symmetry reduction of a generalized symmetry of the Toda system
102(1)
3.2.7 The inhomogeneous Toda lattices
103(3)
3.3 Volterra hierarchy, its symmetries, Backlund transformations, Bianchi identity and continuous limit
106(2)
3.3.1 Backlund transformations
108(1)
3.3.2 Infinite dimensional symmetry algebra
109(2)
3.3.3 Contraction of the symmetry algebras in the continuous limit
111(1)
3.3.4 Symmetry reduction of a generalized symmetry of the Volterra equation
112(1)
3.3.5 Inhomogeneous Volterra equations
113(1)
3.4 Discrete Nonlinear Schrodinger equation, its symmetries, Backlund transformations and continuous limit
113(1)
3.4.1 The dNLS hierarchy and its integrability
114(3)
3.4.2 Lie point symmetries of the dNLS
117(1)
3.4.3 Generalized symmetries of the dNLS
118(2)
3.4.4 Continuous limit of the symmetries of the dNLS
120(1)
3.4.5 Symmetry reductions
121(6)
3.5 The DAE Burgers
127(1)
3.5.1 Backlund transformations for the DAE Burgers and its non linear superposition formula
128(1)
3.5.2 Symmetries for the DAE Burgers
129(1)
4 Integrability of PAEs
129(96)
4.1 Introduction
129(2)
4.2 Discrete time Toda lattice, its hierarchy, symmetries, Backlund transformations and continuous limit
131(1)
4.2.1 Construction of the discrete time Toda lattice hierarchy
131(2)
4.2.2 Isospectral and non isospectral generalized symmetries for the discrete time Toda lattice
133(2)
4.2.3 Symmetry reductions for the discrete time Toda lattice
135(1)
4.2.4 Backlund transformations and symmetries for the discrete time Toda lattice
135(1)
4.3 Discrete time Volterra equation
136(1)
4.3.1 Continuous limit of the discrete time Volterra equation
137(1)
4.3.2 Symmetries for the discrete Volterra equation
137(1)
4.4 Lattice version of the potential KdV, its symmetries and continuous limit
138(1)
4.4.1 Introduction
138(2)
4.4.2 Solution of the discrete spectral problem associated with the IpKdV equation
140(2)
4.4.3 Symmetries of the IpKdV equation
142(3)
4.5 Lattice version of the Schwarzian KdV
145(1)
4.5.1 The integrability of the lSKdV equation
146(1)
4.5.2 Point symmetries of the ISKdV equation
147(1)
4.5.3 Generalized symmetries of the ISKdV equation
148(3)
4.6 Volterra type DAEs and the ABS classification
151(4)
4.6.1 The derivation of the Qv equation
155(1)
4.6.2 Lax pair and Backlund transformations for the ABS equations
156(2)
4.6.3 Symmetries of the ABS equations
158(4)
4.7 Extension of the ABS classification: Boll results
162(2)
4.7.1 Independent equations on a single cell
164(2)
4.7.2 Independent equations on the 2D-lattice
166(2)
4.7.3 Examples
168(7)
4.7.4 The non autonomous Qv equation
175(2)
4.7.5 Symmetries of Boll equations
177(11)
4.7.6 Darboux integrability of trapezoidal H4 and H6 families of lattice equations: first integrals [ 336, 345]
188(8)
4.7.7 Darboux integrability of trapezoidal H4 and H6 families of lattice equations: general solutions [ 336, 344]
196(5)
4.8 Integrable example of quad-graph equations not in the ABS or Boll class
201(2)
4.9 The completely discrete Burgers equation
203(1)
4.10 The discrete Burgers equation from the discrete heat equation
204(1)
4.10.1 Symmetries of the new discrete Burgers
205(3)
4.10.2 Symmetry reduction for the new discrete Burgers equation
208(2)
4.11 Linearization of PAEs through symmetries
210(2)
4.11.1 Examples
212(5)
4.11.2 Necessary and sufficient conditions for a PAE to be linear
217(3)
4.11.3 Four-point linearizable lattice schemes
220(5)
Chapter 3 Symmetries as integrability criteria
225(210)
1 Introduction
225(5)
2 The generalized symmetry method for DΔEs
230(58)
2.1 Generalized symmetries and conservation laws
231(8)
2.2 First integrability condition
239(4)
2.3 Formal symmetries and further integrability conditions
243(8)
2.4 Formal conserved density
251(5)
2.4.1 Why the shape of scalar S-integrable evolutionary DΔEs are symmetric
256(2)
2.4.2 Discussion of PDEs from the point of view of Theorem 34
258(1)
2.4.3 Discussion of PΔEs from the point of view of Theorem 34
259(3)
2.5 Discussion of the integrability conditions
262(1)
2.5.1 Derivation of integrability conditions from the existence of conservation laws
262(1)
2.5.2 Explicit form of the integrability conditions
263(1)
2.5.3 Construction of conservation laws from the integrability conditions
264(1)
2.5.4 Left and right order of generalized symmetries
265(1)
2.6 Hamiltonian equations and their properties
266(3)
2.7 Discrete Miura transformations and master symmetries
269(6)
2.8 Generalized symmetries for systems of lattice equations: Toda type equations
275(6)
2.9 Integrability conditions for relativistic Toda type equations
281(7)
3 Classification results
288(28)
3.1 Volterra type equations
288(1)
3.1.1 Examples of classification
288(4)
3.1.2 Lists of equations, transformations and master symmetries
292(5)
3.2 Toda type equations
297(4)
3.3 Relativistic Toda type equations
301(1)
3.3.1 Non point connection between Lagrangian and Hamiltonian equations, and properties of Lagrangian equations
302(4)
3.3.2 Hamiltonian form of relativistic lattice equations
306(2)
3.3.3 Lagrangian form of relativistic lattice equations
308(2)
3.3.4 Relations between the presented lists of relativistic equations
310(2)
3.3.5 Master symmetries for the relativistic lattice equations
312(4)
4 Explicit dependence on the discrete spatial variable n and time t
316(14)
4.1 Dependence on n in Volterra type equations
316(1)
4.1.1 Discussion of the general theory
316(4)
4.1.2 Examples
320(4)
4.2 Toda type equations with an explicit n and t dependence
324(4)
4.3 Example of relativistic Toda type
328(2)
5 Other types of lattice equations
330(9)
5.1 Scalar evolutionary DΔEs of an arbitrary order
330(5)
5.2 Multi-component DΔEs
335(4)
6 Completely discrete equations
339(21)
6.1 Generalized symmetries for PΔEs and integrability conditions
339(1)
6.1.1 Preliminary definitions
339(3)
6.1.2 Derivation of the first integrability conditions
342(3)
6.1.3 Integrability conditions for five point symmetries
345(5)
6.2 Testing PΔEs for the integrability and some classification results
350(1)
6.2.1 A simple classification problem
350(3)
6.2.2 Further application of the method to examples and classes of equations
353(7)
7 Linearizability through change of variables in PΔEs
360(37)
7.1 Three-point PΔEs linearizable by local and non local transformations
362(1)
7.1.1 Linearizability conditions
363(3)
7.1.2 Classification of complex multilinear equations defined on a three-point lattice linearizable by one-point transformations
366(3)
7.1.3 Linearizability by a Cole--Hopf transformation
369(2)
7.1.4 Classification of complex multilinear equations defined on three points linearizable by Cole--Hopf transformation
371(1)
7.2 Nonlinear equations on a quad-graph linearizable by one-point, two-point and generalized Cole--Hopf transformations
372(1)
7.2.1 Linearization by one-point transformations
372(2)
7.2.2 Two-point transformations
374(5)
7.2.3 Linearization by a generalized Cole--Hopf transformation to an homogeneous linear equation
379(4)
7.2.4 Examples
383(9)
7.3 Results on the classification of multilinear PΔEs linearizable by point transformation on a square lattice
392(1)
7.3.1 Quad-graph PΔEs linearizable by a point transformation
392(2)
7.3.2 Classification of complex autonomous multilinear quad-graph PΔEs linearizable by a point transformation
394(3)
Appendix A Construction of lattice equations and their Lax pair
397(10)
Appendix B Transformation groups for quad lattice equations
407(6)
Appendix C Algebraic entropy of the non autonomous Boll equations
413(8)
1 Algebraic entropy test for H4 and H6 trapezoidal equations
413(3)
2 Algebraic entropy for the non autonomous YdKN equation and its subcases
416(5)
Appendix D Translation from Russian of RI Yamilov, On the classification of discrete equations, reference [ 841]
421(12)
1 Proof of the conditions (D.2--D.4)
422(2)
2 Nonlinear differential difference equations satisfying conditions (D.2--D.4)
424(1)
3 List of non linear differential difference equations of type I satisfying conditions (D.2, D.4)
425(8)
Appendix E No quad-graph equation can have a generalized symmetry given by the Narita-Itoh-Bogoyavlensky equation
433(2)
Bibliography 435(38)
Subject Index 473
Decio Levi, Roma Tre University, Rome, Italy, and INFN, Roma Tre Section, Rome, Italy.

Pavel Winternitz, Universite de Montreal, QC, Canada.

Ravil I. Yamilov, UFA Federal Research Center of the Russian Academy of Science, Russia.