Book contents
- Frontmatter
- Contents
- List of figures
- List of contributors
- Preface
- Introduction
- 1 Lagrangian and Hamiltonian Formalism for Discrete Equations: Symmetries and First Integrals
- 2 Painlevé Equations: Continuous, Discrete and Ultradiscrete
- 3 Definitions and Predictions of Integrability for Difference Equations
- 4 Orthogonal Polynomials, their Recursions, and Functional Equations
- 5 Discrete Painlevé Equations and Orthogonal Polynomials
- 6 Generalized Lie Symmetries for Difference Equations
- 7 Four Lectures on Discrete Systems
- 8 Lectures on Moving Frames
- 9 Lattices of Compact Semisimple Lie Groups
- 10 Lectures on Discrete Differential Geometry
- 11 Symmetry Preserving Discretization of Differential Equations and Lie Point Symmetries of Differential-Difference Equations
- References
6 - Generalized Lie Symmetries for Difference Equations
Published online by Cambridge University Press: 05 July 2011
Book contents
- Frontmatter
- Contents
- List of figures
- List of contributors
- Preface
- Introduction
- 1 Lagrangian and Hamiltonian Formalism for Discrete Equations: Symmetries and First Integrals
- 2 Painlevé Equations: Continuous, Discrete and Ultradiscrete
- 3 Definitions and Predictions of Integrability for Difference Equations
- 4 Orthogonal Polynomials, their Recursions, and Functional Equations
- 5 Discrete Painlevé Equations and Orthogonal Polynomials
- 6 Generalized Lie Symmetries for Difference Equations
- 7 Four Lectures on Discrete Systems
- 8 Lectures on Moving Frames
- 9 Lattices of Compact Semisimple Lie Groups
- 10 Lectures on Discrete Differential Geometry
- 11 Symmetry Preserving Discretization of Differential Equations and Lie Point Symmetries of Differential-Difference Equations
- References
Summary
Introduction
In this chapter we discuss the application of generalized symmetries to the investigation of difference and differential-difference equations. This is a sequel to the presentation of P. Winternitz where Lie point symmetries for difference equations have been introduced and studied in detail. In particular it has been shown there that for a given discrete equation, unless we allow for variable lattices, i.e., we consider a difference scheme, very few symmetries are present. So, if we want to get symmetries for difference equations, either we consider the point symmetries of a difference scheme or we extend the class of symmetries to the case of the generalized symmetries. In the following we will proceed in this second direction and analyze the structure of the generalized symmetries for a difference equation. We will limit ourselves to consider just partial difference equations (with two independent variables) where the lattice is fixed and non-transformable and either all independent variables are discrete (n,m) or one is discrete n and one is continuous t. We will limit our discussion to the case of scalar equations of a low order, i.e., when the dependent variable is a scalar and the differential difference equations involve at most derivatives of the second order of the fields and nearest neighboring interactions.
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- Symmetries and Integrability of Difference Equations , pp. 160 - 190Publisher: Cambridge University PressPrint publication year: 2011
References
[1] , , and 2000. The symmetry approach to the integrability problem. Theoret. and Math. Phys., 125(3), 1603–1661.CrossRefGoogle Scholar
[2] 1991. Why are certain nonlinear PDEs both widely applicable and integrable? Pages 1–62 of: What is integrability? Springer Ser. Nonlinear Dynam. Berlin: Springer.Google Scholar
[3] , 1977. Generating exactly soluble nonlinear discrete evolution equations by a generalized Wronskian technique. J. Mathematical Phys. 18(4), 690–700.CrossRefGoogle Scholar
[4] 1993. A finite-difference analogue of Noether's theorem. Phys. Dokl., 38(2), 66–68.Google Scholar
[5] 1974. On the Toda lattice. II. Inverse-scattering solution. Progr. Theoret. Phys. 51, 703–716.CrossRefGoogle Scholar
[6] 1980. A symmetry approach to exactly solvable evolution equations. J. Math. Phys., 21(6), 1318–1325.CrossRefGoogle Scholar
[7] , and 2004. A variational complex for difference equations. Found. Comput. Math., 4(2), 187–217.CrossRefGoogle Scholar
[8] , and 1953. On the contraction of groups and their representations. Proc. Nat. Acad. Sci. U.S.A., 39, 510–524.CrossRefGoogle ScholarPubMed
[9] 1981. Nonlinear differential-difference equations as Bäcklund transformations. J. Phys. A, 14(5), 1083–1098.CrossRefGoogle Scholar
[10] , and 1980. Bäcklund transformations and nonlinear differential difference equations. Proc. Nat. Acad. Sci. U.S.A., 77(9, part 1), 5025–5027.CrossRefGoogle ScholarPubMed
[11] , and 1997. Conditions for the existence of higher symmetries of evolutionary equations on the lattice. J. Math. Phys., 38(12), 6648–6674.CrossRefGoogle Scholar
[12] , and 2009. The generalized symmetry method for discrete equations. J. Phys. A, 42(45), 454012.CrossRefGoogle Scholar
[13] 2010. Integrability test for discrete equations via generalized symmetries. Symmetries in Nature (AIP Conference Proceedings 1323) ed. , , , (Melville, New York: AIP), 203–214.Google Scholar
[14] , 2011. Generalized symmetry integrability test for discrete equations on the square lattice. J. Phys. A, 44(14) 145207 (22 pp)CrossRefGoogle Scholar
[15] , , , and 2008. On Miura transformations and Volterra-type equations associated with the Adler-Bobenko-Suris equations. SIGMA Symmetry Integrability Geom. Methods Appl., 4, Paper 077.Google Scholar
[16] , 2006. Continuous symmetries of difference equations. J. Phys. A 39(2), R1–R63.CrossRefGoogle Scholar
[17] , , and 1987. A symmetric approach to the classification of nonlinear equations. Complete lists of integrable systems. Russian Math. Surveys, 42(4), 1–63.CrossRefGoogle Scholar
[18] , , and 1991. The symmetry approach to classification of integrable equations. Pages 115–184 of: What is integrability? Springer Ser. Nonlinear Dynam. Berlin: Springer.CrossRefGoogle Scholar
[19] , , and 2010. Recursion operators, conservation laws and integrability conditions for difference equations. arXiv:1004.5346.
[20] 1918. Invariante Variationsprobleme. Nachr. v. d. Ges. d. Wiss. zu Göttingen, 235–257. See Noether E. 1971. Invariant variation problems. Transport Theory Statist. Phys., 1(3), 186–207.Google Scholar
[21] 1993. Applications of Lie groups to Differential Equations. 2nd edn. Grad. Texts in Math., vol. 107. New York: Springer.CrossRefGoogle Scholar
[22] , and 2007. Symmetries of integrable difference equations on the quad-graph. Stud. Appl. Math., 119(3), 253–269.CrossRefGoogle Scholar
[23] 1989. Theory of nonlinear lattices. Second edition. Springer Series in Solid-State Sciences, 20. Springer-Verlag, Berlin, x+225 pp.CrossRefGoogle Scholar
[24] 1980. Conservation laws for the discrete Korteweg–de Vries equation. Comput. Methods Appl. Mech. Engrg., 44, 164–173. in Russian.Google Scholar
[25] 1983. Classification of discrete evolution equations. Uspekhi Mat. Nauk, 38(6), 155–156.Google Scholar
[26] 2006. Symmetries as integrability criteria for differential difference equations. J. Phys. A, 39(45), R541–R623.CrossRefGoogle Scholar
[27] 2007. Integrability conditions for analogues of the relativistic Toda chain. Theoret. and Math. Phys., 151(1), 492–504.CrossRefGoogle Scholar
[28] , and 2004. Integrability conditions for n and t dependent dynamical lattice equations. J. Nonlinear Math. Phys., 11(1), 75–101.CrossRefGoogle Scholar
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