Graded Differential Equations and Their Deformations: A Computational Theory for Recursion Operators

I. S. Krasil’Shchik and P. H. M. Kersten
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I. S. Krasil’Shchik: Moscow Institute for Municipal Economy and Civil Engineering
P. H. M. Kersten: University of Twente, Department of Applied Mathematics

A chapter in Geometric and Algebraic Structures in Differential Equations, 1995, pp 167-191 from Springer

Abstract: Abstract An algebraic model for nonlinear partial differential equations (PDE) in the category of n-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, H ∇ 0 A) generalizes the Lie algebra of symmetries for PDE’s, while H ∇ 1 (A) are identified with equivalence classes of infinitesimal deformations. It is shown that elements of a certain part of H ∇ 1 (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.

Keywords: Nonlinear Partial Differential Equation; Partial Differential Equation; Recursion Operator; Cartan Connection; Flat Connection (search for similar items in EconPapers)
Date: 1995
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DOI: 10.1007/978-94-009-0179-7_11

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