Graded differential equations and their deformations: A computational theory for recursion operators |
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Authors: | I. S. Krasil'shchik P. H. M. Kersten |
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Affiliation: | (1) Moscow Institute for Municipal Economy and Civil Engineering, Moscow, Russia;(2) Department of Applied Mathematics, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands |
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Abstract: | An algebraic model for nonlinear partial differential equations (PDE) in the category ofn-graded modules is constructed. Based on the notion of the graded Frölicher-Nijenhuis bracket, cohomological invariants H*(A) are related to each object (A, ) of the theory. Within this framework, H0 (A) generalizes the Lie algebra of symmetries for PDE's, while H1(A) are identified with equivalence classes of infinitesimal deformations. It is shown that elements of a certain part of H1(A) can be interpreted as recursion operators for the object (A, ), i.e. operators giving rise to infinite series of symmetries. Explicit formulas for computing recursion operators are deduced. The general theory is illustrated by a particular example of a graded differential equation, i.e. the Super KdV equation.Tverskoy-Yamskoy per. 14, Apt. 45, 125047 Moscow, Russia. |
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Keywords: | 58F07 58G07 58H10 58H15 58G37 58A50 35Q53 35Q55 35Q58 58G35 16W55 |
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