Two-Dimensional Dilaton Gravity and Toda - Liouville Integrable Models

摘要

General properties of a class of two-dimensional dilaton gravity (DG)theories with multi-exponential potentials are studied and a subclass of thesetheories, in which the equations of motion reduce to Toda and Liouvilleequations, is treated in detail. A combination of parameters of the equationsshould satisfy a certain constraint that is identified and solved for thegeneral multi-exponential model. From the constraint it follows that in DGtheories the integrable Toda equations, generally, cannot appear withoutaccompanying Liouville equations. We also show how the wave-like solutions of the general Toda-Liouvillesystems can be simply derived. In the dilaton gravity theory, these solutionsdescribe nonlinear waves coupled to gravity as well as static states andcosmologies. A special attention is paid to making the analytic structure ofthe solutions of the Toda equations as simple and transparent as possible, withthe aim to gain a better understanding of realistic theories reduced todimensions 1+1 and 1+0 or 0+1.