Multi - exponential models of (1+1)-dimensional dilaton gravity and Toda - Liouville integrable models

The general properties of a class of two-dimensional dilaton gravity (DG) theories with multiexponential potentials are studied and a subclass of these theories, in which the equations of motion reduce to Toda and Liouville equations, is treated in detail. A combination of parameters of the equations should satisfy a certain constraint that is identified and solved for the general multi-exponential model. From the constraint it follows that in DG theories the integrable Toda equations, generally, cannot appear without accompanying Liouville equations. The most difficult problem in the two-dimensional Toda Liouville DG is to solve the energy and momentum constraints. We discuss this problem using the simplest examples and identify the main obstacles to finding its analytic solution. Then we consider a subclass of integrable two-dimensional theories, in which scalar matter fields satisfy the Toda equations while the two-dimensional metric is trivial; the simplest case is considered in some detail, and on this example we outline how the general solution can be obtained. We also show how the wave-like solutions of the general Toda Liouville systems can be simply derived. In the dilaton gravity theory, these solutions describe nonlinear waves coupled to gravity as well as static states and cosmologies. For static states and cosmologies we propose and study a more general one-dimensional Toda Liouville model typically emerging in one-dimensional reductions of higher-dimensional gravity and supergravity theories. A special attention is paid in this paper to making the analytic structure of the solutions of the Toda equations as simple and transparent as possible, with the aim to gain a better understanding of realistic theories reduced to dimensions 1+1 and 1+0 or 0+1.