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.
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Papageorgiou, V., Grammaticos, B. & Ramani, A. Orthogonal polynomial approach to discrete Lax pairs for initial boundary-value problems of the QD algorithm. Lett Math Phys 34, 91–101 (1995). https://doi.org/10.1007/BF00739089
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DOI: https://doi.org/10.1007/BF00739089