Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 087, 23 pages      arXiv:1211.3803      https://doi.org/10.3842/SIGMA.2012.087

Geometric Theory of the Recursion Operators for the Generalized Zakharov-Shabat System in Pole Gauge on the Algebra sl(n,C)

Alexandar B. Yanovski a and Gaetano Vilasi b
a) Department of Mathematics & Applied Mathematics, University of Cape Town, Rondebosch 7700, Cape Town, South Africa
b) Dipartimento di Fisica, Università degli Studi di Salerno, INFN, Sezione di Napoli-GC Salerno, Via Ponte Don Melillo, 84084, Fisciano (Salerno), Italy

Received May 17, 2012, in final form November 05, 2012; Published online November 16, 2012

Abstract
We consider the recursion operator approach to the soliton equations related to the generalized Zakharov-Shabat system on the algebra sl(n,C) in pole gauge both in the general position and in the presence of reductions. We present the recursion operators and discuss their geometric meaning as conjugate to Nijenhuis tensors for a Poisson-Nijenhuis structure defined on the manifold of potentials.

Key words: Lax representation; recursion operators; Nijenhuis tensors.

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