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Differential Calculi on Associative Algebras and Integrable Systems

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Book cover Algebraic Structures and Applications (SPAS 2017)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 317))

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Abstract

After an introduction to some aspects of bidifferential calculus on associative algebras, we focus on the notion of a “symmetry” of a generalized zero curvature equation and derive Bäcklund and (forward, backward and binary) Darboux transformations from it. We also recall a matrix version of the binary Darboux transformation and, inspired by the so-called Cauchy matrix approach, present an infinite system of equations solved by it. Finally, we sketch recent work on a deformation of the matrix binary Darboux transformation in bidifferential calculus, leading to a treatment of integrable equations with sources.

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Notes

  1. 1.

    Setting alternatively \(A=0\) and keeping B, corresponds to an exchange of \(\mathrm {d}\) and \(\bar{\mathrm {d}}\) in the following equations.

  2. 2.

    This generalizes the notion of a fundamental 1-form introduced in [31] to bidifferential calculus. In the spirit of [24], it may also be called a “conservation law”. Also see Sect. 16.2.5.

  3. 3.

    Here we depart from the previous scheme since \(\tilde{J}_1 = J_1 + \mathrm {d} ( \phi \, \tilde{\chi }_0)\), where \(J_1 = \bar{D} \tilde{\chi }_0\). \(\tilde{J}_1\) is not annihilated by \(\bar{D}\) if \(\phi \ne 0\).

  4. 4.

    A bidifferential calculus for KdV is recalled in Example 16.2 in Sect. 16.6.

  5. 5.

    F M-H had a very illuminating discussion about all this with Martin Bordemann in June 2000.

  6. 6.

    There is a wrong factor in front of the cubic nonlinearities in (4.5) of [7].

  7. 7.

    We are grateful to Dmitry Demskoi and Maxim Pavlov for informing us about this.

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Author information

Authors and Affiliations

  1. Department of Financial and Management Engineering, University of the Aegean, 82100, Chios, Greece

    Aristophanes Dimakis

  2. Max Planck Institute for Dynamics and Self-Organization, 37077, Göttingen, Germany

    Folkert Müller-Hoissen

  3. Institute for Theoretical Physics, University of Göttingen, 37077, Göttingen, Germany

    Folkert Müller-Hoissen

Authors
  1. Aristophanes Dimakis
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Correspondence to Folkert Müller-Hoissen .

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Editors and Affiliations

  1. Division of Applied Mathematics, School of Education, Culture and Communication, Mälardalen University, Västerås, Sweden

    Sergei Silvestrov

  2. Division of Applied Mathematics, School of Education, Culture and Communication, Mälardalen University, Västerås, Sweden

    Anatoliy Malyarenko

  3. Division of Applied Mathematics, School of Education, Culture and Communication, Mälardalen University, Västerås, Sweden

    Milica Rančić

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Dimakis, A., Müller-Hoissen, F. (2020). Differential Calculi on Associative Algebras and Integrable Systems. In: Silvestrov, S., Malyarenko, A., Rančić, M. (eds) Algebraic Structures and Applications. SPAS 2017. Springer Proceedings in Mathematics & Statistics, vol 317. Springer, Cham. https://doi.org/10.1007/978-3-030-41850-2_16

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