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Letters in Mathematical Physics

, Volume 34, Issue 2, pp 91–101 | Cite as

Orthogonal polynomial approach to discrete Lax pairs for initial boundary-value problems of the QD algorithm

  • V. Papageorgiou
  • B. Grammaticos
  • A. Ramani
Article

Abstract

Using orthogonal polynomial theory, we construct the Lax pair for the quotient-difference algorithm in the natural Rutishauser variables. We start by considering the family of orthogonal polynomials corresponding to a given linear form. Shifts on the linear form give rise to adjacent families. A compatible set of linear problems is made up from two relations connecting adjacent and original polynomials. Lax pairs for several initial boundary-value problems are derived and we recover the discrete-time Toda chain equations of Hirota and of Suris. This approach allows us to derive a Bäcklund transform that relates these two different discrete-time Toda systems. We also show that they yield the same bilinear equation up to a gauge transformation. The singularity confinement property is discussed as well.

Mathematics Subject Classifications (1991)

Primary: 58F07, 39A10 Secondary: 65F15, 70F10 

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Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • V. Papageorgiou
    • 1
  • B. Grammaticos
    • 2
  • A. Ramani
    • 3
  1. 1.Institute for Nonlinear StudiesClarkson UniversityPotsdamUSA
  2. 2.LPN, Université Paris VIIParisFrance
  3. 3.CPT, Ecole Polytechnique, CNRS, UPR 14PalaiseauFrance

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