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

In: Geometric and Algebraic Structures in Differential Equations

Author

Listed:
  • I. S. Krasil’Shchik

    (Moscow Institute for Municipal Economy and Civil Engineering)

  • P. H. M. Kersten

    (University of Twente, Department of Applied Mathematics)

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.

Suggested Citation

  • I. S. Krasil’Shchik & P. H. M. Kersten, 1995. "Graded Differential Equations and Their Deformations: A Computational Theory for Recursion Operators," Springer Books, in: P. H. M. Kersten & I. S. Krasil’Shchik (ed.), Geometric and Algebraic Structures in Differential Equations, pages 167-191, Springer.
    • Handle: RePEc:spr:sprchp:978-94-009-0179-7_11
    DOI: 10.1007/978-94-009-0179-7_11
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