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.
Author(s): Decio Levi, Pavel Winternitz, Ravil I. Yamilov
Series: CRM Monograph Series
Publisher: American Mathematical Society
Year: 2023
Language: English
Pages: 518
City: Providence
Front Cover
Contents
Foreword
List of Figures
List of Tables
Preface
Acknowledgment
Chapter 1. Introduction
1. Lie point symmetries of differential equations, their extensions and applications
2. What is a lattice
2.1. 1-dimensional lattices
2.2. 2-dimensional lattices
2.3. Differential and difference operators on the lattice
2.4. Grids and lattices in the description of difference equations
2.4.1. Cartesian lattices
2.4.2. Galilei invariant lattice
2.4.3. Exponential lattice
2.4.4. Polar coordinate systems
2.5. Clairaut–Schwarz–Young theorem on the lattices and its consequences
2.5.1. Commutativity and non commutativity of difference operators
3. What is a difference equation
3.1. Examples
4. How do we find symmetries for difference equations
4.1. Examples
4.1.1. Lie point symmetries of the discrete time Toda lattice
4.1.2. Lie point symmetries of DΔEs
4.1.3. Lie point symmetries of the Toda lattice
4.1.4. Classification of DΔEs
4.1.5. Lie point symmetries of the two dimensional Toda equation
4.2. Lie point symmetries preserving discretization of ODEs
4.3. Group classification and solution of OΔEs
4.3.1. Symmetries of second order ODEs
4.3.2. Symmetries of the three-point difference schemes
4.3.3. \sloppy Lagrangian formalism and solutions of three-point OΔS
5. What we leave out on symmetries in this book
6. Outline of the book
Chapter 2. Integrability and symmetries of nonlinear differential and difference equations in two independent variables
1. Introduction
2. Integrability of PDEs
2.1. Introduction
2.2. All you ever wanted to know about the integrability of the KdV equation and its hierarchy
2.2.1. The KdV hierarchy: recursion operator
2.2.2. The Bäcklund transformations, Darboux operators and Bianchi identity for the KdV hierarchy
2.2.3. The conservations laws for the KdV equation
2.2.4. The symmetries of the KdV hierarchy
2.2.5. Lie algebra of the symmetries
2.2.6. Relation between Bäcklund transformations and isospectral symmetries
2.2.7. Symmetry reductions of the KdV equation
2.3. The cylindrical KdV, its hierarchy and Darboux and Bäcklund transformations
2.4. Integrable PDEs as infinite-dimensional superintegrable systems
2.5. Integrability of the Burgers equation, the prototype of linearizable PDEs
2.5.1. Bäcklund transformation and Bianchi identity for the Burgers hierarchy of equations
2.5.2. Symmetries of the Burgers equation
2.5.3. Symmetry reduction by Lie point symmetries
2.6. General ideas on linearization
2.6.1. Linearization of PDEs through symmetries
3. Integrability of DΔEs
3.1. Introduction
3.2. The Toda lattice, the Toda system, the Toda hierarchy and their symmetries
3.2.1. Symmetries for the Toda hierarchy
3.2.2. The Lie algebra of the symmetries for the Toda system and Toda lattice
3.2.3. Contraction of the symmetry algebras in the continuous limit
3.2.4. Bäcklund transformations and Bianchi identities for the Toda system and Toda lattice
3.2.5. Relation between Bäcklund transformations and isospectral symmetries
3.2.6. Symmetry reduction of a generalized symmetry of the Toda system
3.2.7. The inhomogeneous Toda lattices
3.3. Volterra hierarchy, its symmetries, Bäcklund transformations, Bianchi identity and continuous limit
3.3.1. Bäcklund transformations
3.3.2. Infinite dimensional symmetry algebra
3.3.3. Contraction of the symmetry algebras in the continuous limit
3.3.4. Symmetry reduction of a generalized symmetry of the Volterra equation
3.3.5. Inhomogeneous Volterra equations
3.4. Discrete Nonlinear Schrödinger equation, its symmetries, Bäcklund transformations and continuous limit
3.4.1. The dNLS hierarchy and its integrability
3.4.2. Lie point symmetries of the dNLS
3.4.3. Generalized symmetries of the dNLS
3.4.4. Continuous limit of the symmetries of the dNLS
3.4.5. Symmetry reductions
3.5. The DΔE Burgers
3.5.1. Bäcklund transformations for the DΔE Burgers and its non linear superposition formula
3.5.2. Symmetries for the DΔE Burgers
4. Integrability of PΔEs
4.1. Introduction
4.2. Discrete time Toda lattice, its hierarchy, symmetries, Bäcklund transformations and continuous limit
4.2.1. Construction of the discrete time Toda lattice hierarchy
4.2.2. Isospectral and non isospectral generalized symmetries for the discrete time Toda lattice
4.2.3. Symmetry reductions for the discrete time Toda lattice.
4.2.4. Bäcklund transformations and symmetries for the discrete time Toda lattice.
4.3. Discrete time Volterra equation
4.3.1. Continuous limit of the discrete time Volterra equation
4.3.2. Symmetries for the discrete Volterra equation
4.4. Lattice version of the potential KdV, its symmetries and continuous limit
4.4.1. Introduction
4.4.2. Solution of the discrete spectral problem associated with the lpKdV equation
4.4.3. Symmetries of the lpKdV equation
4.5. Lattice version of the Schwarzian KdV
4.5.1. The integrability of the lSKdV equation
4.5.2. Point symmetries of the lSKdV equation
4.5.3. Generalized symmetries of the lSKdV equation
4.6. Volterra type DΔEs and the ABS classification
4.6.1. The derivation of the ?_{?} equation
4.6.2. Lax pair and Bäcklund transformations for the ABS equations
4.6.3. Symmetries of the ABS equations
4.7. Extension of the ABS classification: Boll results.
4.7.1. Independent equations on a single cell
4.7.2. Independent equations on the 2?-lattice
4.7.3. Examples
4.7.4. The non autonomous \QV equation
4.7.5. Symmetries of Boll equations
4.7.6. Darboux integrability of trapezoidal ?⁴ and ?⁶ families of lattice equations: first integrals [336, 345]
4.7.7. Darboux integrability of trapezoidal ?⁴ and ?⁶ families of lattice equations: general solutions [336, 344]
4.8. Integrable example of quad-graph equations not in the ABS or Boll class
4.9. The completely discrete Burgers equation
4.10. The discrete Burgers equation from the discrete heat equation
4.10.1. Symmetries of the new discrete Burgers
4.10.2. Symmetry reduction for the new discrete Burgers equation
4.11. Linearization of PΔEs through symmetries
4.11.1. Examples.
4.11.2. Necessary and sufficient conditions for a PΔE to be linear.
4.11.3. Four-point linearizable lattice schemes
Chapter 3. Symmetries as integrability criteria
1. Introduction
2. The generalized symmetry method for DΔEs
2.1. Generalized symmetries and conservation laws
2.2. First integrability condition
2.3. Formal symmetries and further integrability conditions
2.4. Formal conserved density
2.4.1. Why the shape of scalar S-integrable evolutionary DΔEs are symmetric
2.4.2. Discussion of PDEs from the point of view of Theorem 34
2.4.3. Discussion of PΔEs from the point of view of Theorem 34
2.5. Discussion of the integrability conditions
2.5.1. Derivation of integrability conditions from the existence of conservation laws
2.5.2. Explicit form of the integrability conditions
2.5.3. Construction of conservation laws from the integrability conditions
2.5.4. Left and right order of generalized symmetries
2.6. Hamiltonian equations and their properties
2.7. Discrete Miura transformations and master symmetries
2.8. Generalized symmetries for systems of lattice equations: Toda type equations
2.9. Integrability conditions for relativistic Toda type equations
3. Classification results
3.1. Volterra type equations
3.1.1. Examples of classification
3.1.2. Lists of equations, transformations and master symmetries
3.2. Toda type equations
3.3. Relativistic Toda type equations
3.3.1. Non point connection between Lagrangian and Hamiltonian equations, and properties of Lagrangian equations
3.3.2. Hamiltonian form of relativistic lattice equations
3.3.3. Lagrangian form of relativistic lattice equations
3.3.4. Relations between the presented lists of relativistic equations
3.3.5. Master symmetries for the relativistic lattice equations
4. Explicit dependence on the discrete spatial variable ? and time ?
4.1. Dependence on ? in Volterra type equations
4.1.1. Discussion of the general theory
4.1.2. Examples
4.2. Toda type equations with an explicit ? and ? dependence
4.3. Example of relativistic Toda type
5. Other types of lattice equations
5.1. Scalar evolutionary DΔEs of an arbitrary order
5.2. Multi-component DΔEs
6. Completely discrete equations
6.1. Generalized symmetries for PΔEs and integrability conditions
6.1.1. Preliminary definitions
6.1.2. Derivation of the first integrability conditions
6.1.3. Integrability conditions for five point symmetries
6.2. Testing PΔEs for the integrability and some classification results
6.2.1. A simple classification problem
6.2.2. Further application of the method to examples and classes of equations
7. Linearizability through change of variables in PΔEs
7.1. Three-point PΔEs linearizable by local and non local transformations
7.1.1. Linearizability conditions.
7.1.2. Classification of complex multilinear equations defined on a three-point lattice linearizable by one-point transformations
7.1.3. Linearizability by a Cole–Hopf transformation
7.1.4. Classification of complex multilinear equations defined on three points linearizable by Cole-Hopf transformation
7.2. Nonlinear equations on a quad-graph linearizable by one-point, two-point and generalized Cole–Hopf transformations
7.2.1. Linearization by one-point transformations
7.2.2. Two-point transformations
7.2.3. Linearization by a generalized Cole–Hopf transformation to an homogeneous linear equation
7.2.4. Examples
7.3. Results on the classification of multilinear PΔEs linearizable by point transformation on a square lattice
7.3.1. Quad-graph PΔEs linearizable by a point transformation.
7.3.2. Classification of complex autonomous multilinear quad-graph PΔEs linearizable by a point transformation.
Appendix A. Construction of lattice equations and their Lax pair
Appendix B. Transformation groups for quad lattice equations.
Appendix C. Algebraic entropy of the non autonomous Boll equations
1. Algebraic entropy test for ?⁴ and ?⁶ trapezoidal equations
2. Algebraic entropy for the non autonomous YdKN equation and its subcases.
Appendix D. Translation from Russian of R I Yamilov, On the classification of discrete equations, reference [841].
1. Proof of the conditions (D.2–D.4).
2. Nonlinear differential difference equations satisfying conditions (D.2–D.4).
3. List of non linear differential difference equations of type I satisfying conditions (D.2, D.4).
Appendix E. No quad-graph equation can have a generalized symmetry given by the Narita-Itoh-Bogoyavlensky equation
Bibliography
Index
Back Cover
