Multi-field Systems of the Relativistic Toda Type

Abstract

The multi-field analog of the Toda lattice TL_m+1 studied in the previous chapter, admits also a relativistic generalization, or, better, two different generalizations. Indeed, in Chapter 6 we introduced two different Lax representations of the RTL(a) hierarchy. The multi-field analogs of these Lax representations do not lead to one and the same result. Rather, we get two different multi-field relativistic Toda hierarchies. The first construction is an application of the general ansatz from Sections 6.2-6.4. The hierarchy appearing on this way contains a system which is a direct generalization of the flow RTL_ (a):

$$\begin{gathered} {{{\dot{b}}}_{k}} = \frac{{a_{k}^{{\left( 1 \right)}}}}{{1 + a{{b}_{{k + 1}}}}} - \frac{{a_{{k - 1}}^{{\left( 1 \right)}}}}{{1 + a{{b}_{{k - 1}}}}}, \hfill \\ \dot{a}_{k}^{{\left( j \right)}} = a_{k}^{{\left( j \right)}}\left( {\frac{{{{b}_{{k + j}}}}}{{1 + \alpha {{b}_{{k + j}}}}} - \frac{{{{b}_{k}}}}{{1 + \alpha {{b}_{k}}}}} \right) + \left( {\frac{{a_{k}^{{\left( {j + 1} \right)}}}}{{1 + \alpha {{b}_{{k + j + 1}}}}} - \frac{{a_{{k - 1}}^{{\left( {j + 1} \right)}}}}{{1 + \alpha {{b}_{{k - 1}}}}}} \right). \hfill \\ \end{gathered} $$

(13.1.1)