Discrete dressing transformations and Painlevé equations. (English) Zbl 0969.39501
Summary: The authors propose a discrete analog of the dressing transformation. The starting point is a variant of the quotient-difference algorithm which, in this case, corresponds to a linear problem with shifts in the eigenvalues. The proper periodicity conditions lead to one-dimensional systems which are discrete Painlevé equations. The authors obtain thus the alternate \(d-P_{\text{II}}\) equation and a novel form for the discrete \(P_{\text{IV}}\) equation.
MSC:
| 39A12 | Discrete version of topics in analysis |
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
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