Bogoyavlensky Lattices

Abstract

There are three basic families of integrable lattice systems carrying the (somewhat historically incorrect) name ofBogoyavlensky lattices.These systems are enumerated by integer parameters m, p ≥ 1 (p > 1 for the third one) and read:

$$ {\dot a_k} = {a_k}\left( {\sum\limits_{j = 1}^m {{a_{k + j}} - \sum\limits_{j = 1}^m {{a_{k - j}}} } } \right),$$

(17.1.1)

$$ {\dot a_k} = {a_k}\left( {\prod\limits_{j = 1}^p {{a_k} - \prod\limits_{j = 1}^p {{a_{k - j}}} } } \right),$$

(17.1.2)

$$ \dot a_k = \prod\limits_{j = 1}^{p - 1} {a_{k + j}^{ - 1} } - \prod\limits_{j = 1}^{p - 1} {a_{k - j}^{ - 1} } . $$

(17.1.3)

We shall call these systems BLl(m), BL2(p), and BL3(p), respectively. The lattices BLl (m) and BL2(p) are two different generalizations of the Volterra lattice (4.1.1). Indeed the Volterra lattice is the particular case of both of them, corresponding to m = 1 and p = 1, respectively. The lattice BL3(p) after the change of variables ${a_k} \mapsto {c_k} = a_k^{ - 1}$ and $t \mapsto - t$ turns into

$${{\dot{c}}_{k}} = c_{k}^{2}\left( {\prod\limits_{{j = 1}}^{{p - 1}} {{{c}_{{k + j}}}} - \prod\limits_{{j = 1}}^{{p - 1}} {{{c}_{{k - j}}}} } \right). $$

(17.1.4)

This is a generalization of the modified Volterra lattice, more precisely, its particular case (4.14.1), which appears from the above system by p = 2. We shall see that also in general (17.1. 4) is a particular case of the modified BL1( m ), if p = m + 1.