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Hermitian and Pseudo-Hermitian Reduction of the GMV Auxiliary System. Spectral Properties of the Recursion Operators

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Advanced Computing in Industrial Mathematics (BGSIAM 2017)

Part of the book series: Studies in Computational Intelligence ((SCI,volume 793))

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Abstract

We consider simultaneously two different reductions of a Zakharov-Shabat’s spectral problem in pole gauge. Using the concept of gauge equivalence, we construct expansions over the eigenfunctions of the Recursion Operators related to the afore-mentioned spectral problem with arbitrary constant asymptotic values of the potential functions. In doing this, we take into account the discrete spectrum of the scattering operator. Having in mind the applications to the theory of the soliton equations associated to the GMV systems, we show how these expansions modify depending on the symmetries of the functions we expand.

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Notes

  1. 1.

    A more general system was derived independently by Golubchik and Sokolov [9].

  2. 2.

    In [3, 4] FAS have mistakenly been claimed to satisfy \(H\tilde{\chi }^{\pm }(x,\lambda )H = \tilde{\chi }^{\mp }(x,-\lambda )\) .

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Acknowledgements

The work has been supported by the NRF incentive grant of South Africa and grant DN 02–5 of Bulgarian Fund “Scientific Research”.

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Authors and Affiliations

  1. Department of Mathematics and Applied Mathematics, University of Cape Town, Cape Town, South Africa

    A. B. Yanovski

  2. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113, Sofia, Bulgaria

    T. I. Valchev

Authors
  1. A. B. Yanovski
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  2. T. I. Valchev
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Correspondence to A. B. Yanovski .

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Editors and Affiliations

  1. Institute of Information and Communication Technology, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Krassimir Georgiev

  2. Faculty of Applied Mathematics and Informatics, Technical University of Sofia, Sofia, Bulgaria

    Michail Todorov

  3. Institute of Mathematics and Informatics and Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Ivan Georgiev

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Yanovski, A.B., Valchev, T.I. (2019). Hermitian and Pseudo-Hermitian Reduction of the GMV Auxiliary System. Spectral Properties of the Recursion Operators. In: Georgiev, K., Todorov, M., Georgiev, I. (eds) Advanced Computing in Industrial Mathematics. BGSIAM 2017. Studies in Computational Intelligence, vol 793. Springer, Cham. https://doi.org/10.1007/978-3-319-97277-0_35

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