Abstract
These \(\mathcal {C}\)-differential operators are somewhat dual (see Remark 11.1 below) to recursion operators for symmetries considered in Sect. 7. They send cosymmetries of an equation \(\mathcal {E}\) to themselves. (Actually, recursion operators for cosymmetries take solutions of the equation \(\tilde {\ell }_{\mathcal {E}}^*(\psi )=0\) in some covering over \(\mathcal {E}\), i.e. shadows of cosymmetries, to objects of the same nature.) Though these operators are not so popular in applications as their symmetry counterparts, we expose briefly our approach to compute them. In this chapter we give the solution to Problems 1.27 and 1.28.
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Notes
- 1.
As we noted already, mathematical folklore says that an equation admits a local recursion operator only if it is linear with constant coefficients, but we never saw a rigorous and more or less general proof.
References
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Krasil’shchik, J., Verbovetsky, A., Vitolo, R. (2017). Recursion Operators for Cosymmetries. In: The Symbolic Computation of Integrability Structures for Partial Differential Equations. Texts & Monographs in Symbolic Computation. Springer, Cham. https://doi.org/10.1007/978-3-319-71655-8_11
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