Integrable Discrete Equations Derived by Similarity Reduction of the Extended Discrete KP Hierarchy

Details Integrable Discrete Equations Derived by Similarity Reduction of the Extended Discrete KP Hierarchy Integrable Discrete Equations Derived by Similarity Reduction of the Extended Discrete KP Hierarchy

2005-07-03

arxiv.org/abs/nlin/0507004v4

SIGMA 2 (2006), 005, 11 pages

Nonlinear Sciences

Exactly Solvable and Integrable Systems

Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/

Scientific paper

10.3842/SIGMA.2006.005

We consider the extended discrete KP hierarchy and show that similarity reduction of its subhierarchies lead to purely discrete equations with dependence on some number of parameters together with equations governing deformations with respect to these parameters. It is written down discrete equations which naturally generalize the first discrete Painlev\'e equation $\mathrm{dP}_{\rm I}$ in a sense that autonomous version of these equations admit the limit to the first Painlev\'e equation. It is shown that each of these equations describes B\"acklund transformations of Veselov-Shabat periodic dressing lattices with odd period known also as Noumi-Yamada systems of type $A_{2(n-1)}^{(1)}$.

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