Integral Equations and Operator Theory

, Volume 41, Issue 4, pp 472–496 | Cite as

Bundle bispectrality for matrix differential equations

  • Alexander Sakhnovich
  • Jorge P. Zubelli
Article

Abstract

We consider the fundamental solutions of a wide class of first order systems with polynomial dependence on the spectral parameter and rational matrix potentials. Such matrix potentials are rational solutions of a large class of integrable nonlinear equations, which play an important role in different mathematical physics problems. The concept of bispectrality, which was originally introduced by Grünbaum, is extended in a natural way for the systems under consideration and their bispectrality is derived via the representation of the fundamental solutions. This bispectrality is preserved under the flows of the corresponding integrable nonlinear equations. For the case of Dirac type (canonical) systems the complete characterization of the bispectral potentials under consideration is obtained in terms of the system's spectral function.

AMS Classification Primary

37K10 Secondary 34L40 35Q35 35Q58 37K15 

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Copyright information

© Birkhäuser Verlag 2001

Authors and Affiliations

  • Alexander Sakhnovich
    • 1
  • Jorge P. Zubelli
    • 3
  1. 1.University of GlamorganPontypriddUK
  2. 2.Branch of HydroacousticsNational Academy of Sciences of Ukraine, Marine Institute of Hydrophysics, NASUOdessaUkraine
  3. 3.IMPARio de JaneiroBrazil

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