Abstract
It is shown that the methods of Lie group theory, developed for analyzing differential equations, provide equally powerful tools for solving differential-difference equations.
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References
P.J. Olver (1986) Applications of Lie Groups to Differential Equations, Springer, New York.
P. Winternitz (1993) Lie Groups and Solutions of Nonlinear Partial Differential Equations, Integrable systems, Quantum Groups and Quantum Field Theories, Kluwer, Dordrecht, (L.A. Ibort and M.A. Rodriguez, eds)., pp. 429–495.
D. Levi and P. Winternitz (1991) Continuous Symmetries of Discrete Equations, Phys. Lett. A 152, pp. 335–338.
D. Levi and P. Winternitz (1993) Symmetries and Conditional Symmetries of Differential Difference Equations, J. Math. Phys. 34, pp. 3713–3730.
S. Maeda (1980) Canonical Structure and Symmetries for Discrete Systems, Math. Japon. 25, pp. 405–420
S. Maeda (1987) The Similarity Method for Discrete Equations, IMA J. Appl. Math, 38, pp. 129–134.
G.R.W. Quispel, H.W. Capel and R. Sahadevan (1992) Continuous Symmetries of Difference Equations; the Kac-van Moerbeke Equation and Painlevé Reduction, Phys. Lett. 170A, pp. 379–383.
G.W. Bluman and J.D. Cole (1969) The General Similarity Solution of the Heat Equation, J. Math. Mech. 18, pp. 1025–1042.
P.J. Olver and Ph. Rosenau (1987) Group Invariant Solutions of Differential Equations, SIAM J. Appl. Math. 47, pp. 263–278.
D. Levi and P. Winternitz (1989) Nonclassical Symmetry Reduction: Example of the Boussinesq Equation, J. Phys. A: Math. Gen., 22, pp. 2915–2924.
D. David, D. Levi, and P. Winternitz (1988) Equations Invariant under the Symmetry Group of the Kadomtsev-Petviashvili Equation, Phys. Lett. A 129, pp. 161–164.
G. Rideau and P. Winternitz (1990) Nonlinear Equations Invariant under Poincaré, Similitude and Conformal Groups in Two-Dimensional Space-Time, J. Math. Phys 31, pp. 1095–1105.
G. Rideau and P. Winternitz (1993) Invariant Nonlinear Schrödinger Equations in Two-Dimensional Space-Time, J. Math. Phys. 34, pp. 558–569.
J.P. Gazeau and P. Winternitz (1992) Symmetries of Variable Coefficient Kortewegde Vries Equations, J. Math. Phys. 33, pp. 4087–4102.
A.V. Mikhailov (1979) Integrability of a Two Dimensional Generalization of the Toda Chain, JETP Letters 30 pp. 414–418.
A.P. Fordy and J. Gibbons (1980) Integrable Nonlinear Klein-Gordon Equations and Toda Lattices, Commun. Math. Phys. 77, pp. 21–30.
D. David, N. Kamran, D. Levi, and P. Winternitz (1985) Subalgebras of Loop Algebras and Symmetries of the Kadomtsev-Petviashvii Equation, Phys. Rev. Lett. 55, pp. 2111–2113.
B. Champagne and P. Winternitz (1988) On the Infinite Dimensional Symmetry Group of the Davey Stewartson Equations, J. Math. Phys. 29, pp. 1–8.
L. Martina and P. Winternitz (1989) Analysis and Applications of the Symmetry Group of the Multidimensional Three Wave Resonant Interaction Problem, Ann. Phys. 196, pp. 231–277.
A. Yu. Orlov and P. Winternitz (1994) Loop Algebra Symmetries and Commuting Flows for the Kadomtsev-Petviashvili Hierarchy, Preprint CRM-1936, Montréal.
G. Gasper and M. Rahman (1990) Basic Hypergeomnetric Series, Cambridge Univ. Press, Cambridge.
R. Floreanini and L. Vinet (1993) Quantum Algebras and q-Special Functions, Ann. Phys. 221, pp. 53–79.
E.G. Kalnins, H.L. Manocha and W. Miller Jr. (1992) Models of q-algebra Representations: Tensor Products of Special Unitary and Oscillator Algebras, J. Math. Phys. 33 pp. 2365–2383.
G. Rideau and P. Winternitz (1993) Representations of the Quantum Algebra suq (2) on a Real Two-Dimensional Sphere, J. Math. Phys. 34, pp 6030–6044.
D. Levi and P. Winternitz (1995) Symmetry Classification of Discrete Dynamical Systems, Preprint CRM-2256, Montréal.
R. Floreanini and L. Vinet (1994) Symmetries of q-Difference Heat Equation, Lett. Math. Phys. 32, pp. 37–44.
V.A. Dorodnitsyn (1994) Symmetry of Finite Difference Equations, CRC Handbook of Lie Group Analysis of Differential Equations, Vol. 1 CRC Press, Boca Raton (N.H. Ibragimov, ed).
V.A. Dorodnitsyn, M. Sanielevici, and P. Winternitz (1995) Symmetry Preserving Discretization of Variable Coefficient KdV Equations, Preprint CRM-2230, Montréal.
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Winternitz, P., Levi, D. (1995). Continuous Symmetries of Differential-Difference Equations. In: Bertrand, J., Flato, M., Gazeau, JP., Irac-Astaud, M., Sternheimer, D. (eds) Modern Group Theoretical Methods in Physics. Mathematical Physics Studies, vol 18. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8543-9_25
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DOI: https://doi.org/10.1007/978-94-015-8543-9_25
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