Multiexponential models of (1+1)-dimensional dilaton gravity and Toda-Liouville integrable models |
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Authors: | V de Alfaro A T Filippov |
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Institution: | 1.Dipartimento di Fisica Teorica, INFN,Accademia Scienze,Torino,Italy;2.Joint Institute for Nuclear Research,Dubna,Moscow Oblast,Russia |
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Abstract: | We study general properties of a class of two-dimensional dilaton gravity (DG) theories with potentials containing several
exponential terms. We isolate and thoroughly study a subclass of such theories in which the equations of motion reduce to
Toda and Liouville equations. We show that the equation parameters must satisfy a certain constraint, which we find and solve
for the most general multiexponential model. It follows from the constraint that integrable Toda equations in DG theories
generally cannot appear without accompanying Liouville equations. The most difficult problem in the two-dimensional Toda-Liouville
(TL) DG is to solve the energy and momentum constraints. We discuss this problem using the simplest examples and identify
the main obstacles to solving it analytically. We then consider a subclass of integrable two-dimensional theories where scalar
matter fields satisfy the Toda equations and the two-dimensional metric is trivial. We consider the simplest case in some
detail. In this example, we show how to obtain the general solution. We also show how to simply derive wavelike solutions
of general TL systems. In the DG theory, these solutions describe nonlinear waves coupled to gravity and also static states
and cosmologies. For static states and cosmologies, we propose and study a more general one-dimensional TL model typically
emerging in one-dimensional reductions of higher-dimensional gravity and supergravity theories. We especially attend to making
the analytic structure of the solutions of the Toda equations as simple and transparent as possible. |
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