Recursion operators admitted by non-abelian Burgers equations: some remarks. (English) Zbl 1540.35337
Summary: The recursion operators admitted by different operator Burgers equations, in the framework of the study of nonlinear evolution equations, are here considered. Specifically, evolution equations wherein the unknown is an operator acting on a Banach space are investigated. Here, the mirror non-Abelian Burgers equation is considered: it can be written as \(r_t = r_{x x} + 2 r_x r\). The structural properties of the admitted recursion operator are studied; thus, it is proved to be a strong symmetry for the mirror non-Abelian Burgers equation as well as to be hereditary. These results are proved via direct computations as well as via computer assisted manipulations; ad hoc routines are needed to treat non-Abelian quantities and relations among them. The obtained recursion operator generates the mirror non-Abelian Burgers hierarchy. The latter, when the unknown operator \(r\) is replaced by a real valued function reduces to the usual (commutative) Burgers hierarchy. Accordingly, also the recursion operator reduces to the usual Burgers one.
MSC:
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
Keywords:
recursion operator; Bäcklund transformations; non-abelian nonlinear evolution equations; Burgers equationSoftware:
PDERecursionOperatorReferences:
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