Relativistic Volterra Lattice

Abstract

The previous chapter was devoted to the study of the relativistic Toda lattice. We traced several points of the similarity between this system and the usual Toda lattice. However, one such point remained untouched, and we turn our attention to it now. The usual Toda lattice (TL) is Miura related to the two-field Volterra lattice (VL). It turns out that the same Miura transformations relate the flows RTL±(α) to certain systems which are one-parameter perturbations of VL, and can therefore be called relativistic Volterra lattices and denoted RVL±(α), see Section 7.2. An interesting phenomenon here is that each one of the flows RTL+(α), RTL_(α), being pulled back under two companion Miura transformations, results in two slightly different systems, so that actually each one of the systems RVL+(α), RVL_(α) splits into two versions. We give here those ones of RVL±(α) which are related to the Miura map M2. The flow RVL+(α), i.e., pull-back of RTL+(α) under M2, reads:

$$\left\{ {\begin{array}{*{20}{c}} {{{{\dot{u}}}_{k}} = {{u}_{k}}\left( {{{v}_{k}} - {{v}_{{k - 1}}} + \alpha {{u}_{{k + 1}}}{{v}_{k}} - \alpha {{u}_{k}}{{v}_{{k - 1}}}} \right),} \\ {{{{\dot{v}}}_{k}} = {{v}_{k}}\left( {{{u}_{{k + 1}}} - {{u}_{k}} + \alpha {{u}_{{k + 1}}}{{v}_{k}} - \alpha {{u}_{k}}{{v}_{{k - 1}}}} \right).} \\ \end{array} } \right. $$

(7.1.1)