Explicit Discretizations for Toda-type Systems

Abstract

Recall that the maps dRTL₊(α), dRVL₊(α) introduced in Chapter 6, 7, depend actually on two parameters: the “relativistic” parameter a (entering into the notations explicitly) and the time step h (suppressed from the notations). If one takes an attentive look at the recurrent relations for the auxiliary quantities participating in the definitions of the maps dRTL₊(α), dRVL₊(α) etc., one sees that there exists a special value of the relativistic parameter a, for which these recurrent relations simplify dramatically. This is the value a = h. More precisely, for this value the above-mentioned relations cease to be recurrent. Instead, they immediately deliver closed expressions for the auxiliary variables. This allows eventually to put the equations of motion of the maps under consideration into a closed, and moreover, an explicit form. The same conclusions hold for the maps dRTL_ (α), dRVL_ (α) etc., with a = -h. Since the resulting simplified systems are basically the same (up to the change of time direction), we concentrate on the first case.