Acta Applicandae Mathematicae

, Volume 160, Issue 1, pp 129–167 | Cite as

On the (Non)Removability of Spectral Parameters in \(\mathbb{Z}_{2}\)-Graded Zero-Curvature Representations and Its Applications

  • Arthemy V. Kiselev
  • Andrey O. KrutovEmail author


We generalise to the \(\mathbb{Z}_{2}\)-graded set-up a practical method for inspecting the (non)removability of parameters in zero-curvature representations for partial differential equations (PDEs) under the action of smooth families of gauge transformations. We illustrate the generation and elimination of parameters in the flat structures over \(\mathbb{Z}_{2}\)-graded PDEs by analysing the link between deformation of zero-curvature representations via infinitesimal gauge transformations and, on the other hand, propagation of linear coverings over PDEs using the Frölicher–Nijenhuis bracket.


Zero-curvature representation Spectral parameter Removability Supersymmetry Korteweg–de Vries equation Gardner’s deformation Frölicher–Nijenhuis bracket 

Mathematics Subject Classification (2010)

35Q53 37K25 58J72 58A50 



The authors are grateful to I.S. Krasil’shchik, D.A. Leites, M. Marvan, M.A. Nesterenko, P.J. Olver, W.M. Seiler, and A.M. Verbovetsky for helpful correspondence and constructive criticisms. The authors thank P. Mathieu for his attention to this work; the authors are grateful to the anonymous referees for remarks and advice.

This research was done in part while the first author was visiting at the MPIM (Bonn) and the second author was visiting at Utrecht University and New York University Abu Dhabi; the hospitality and support of these institutions are gratefully acknowledged. The research of the first author was partially supported by JBI RUG project 106552 (Groningen); the second author was supported by ISPU scholarship for young scientists and WCMCS post-doctoral fellowship.


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Authors and Affiliations

  1. 1.Bernoulli Institute for Mathematics, Computer Science and Artificial IntelligenceUniversity of GroningenGroningenThe Netherlands
  2. 2.Institut des Hautes Études Scientifiques (IHÉS)Bures-sur-YvetteFrance
  3. 3.Institute of MathematicsPolish Academy of ScienceWarsawPoland
  4. 4.Independent University of MoscowMoscowRussia

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