On reduction of a two-dimensional generalized Toda lattice to an ordinary differential equations system

Abstract

Reduction of a two-dimensional generalized Toda lattice to an ordinary differential equations system, which defines the functional dependence of Toda functions on the corresponding separable solutions, is given. It is shown, in particular, that between all the equations of the form ∂2ρ/∂x ∂t=φ(∂) only Liouville (L), sine-Gordon (SG) and Bullough-Dodd (BD) equations, which are associated with simple Lie algebras of a finite growth of rank 1, lead to the third Painleve equation (P3). The last circumstance allows one, probably, to assume the existence of a deep relation between the complete integrability condition for the corresponding class of dynamical systems and the criterion of the absence of movable critical points in the solutions of an ordinary differential equation system of the second order. If this is so, it means that Painlee's equations and transcendents can be generalized for multi-component cases.