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

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 TodaLiouville (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 twodimensional 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.

Solving dynamical equations in general homogeneous isotropic cosmologies with a scalaron

“…In this paper, we consider the reduction of the two-dimensional theory to equations describing isotropic cosmology and ignore other one-dimensional reductions studied in our previous work [20] - [25]. The procedure and notation is briefly described in Appendix 6.1.…”

“…The structure of the spherically symmetric reduction, which is the two-dimensional field theory, allows one to find some integrable classes of models if we make strong simplifying assumptions about their potentials. For some multi-exponential potentials and the simplest ('minimal') coupling of scalars to gravity, there exist integrable systems related to Liouville or Toda-Liouville two-dimensional theories (see [20] - [25]). 4 For the one-dimensional cosmological reductions with one scalaron there might exist more integrable models.…”

“…The explicit representations for ψ(α) and α(ψ) can easily be derived in the asymptotic region t → ∞ 9. More complex exponential integrable models with one and more scalar fields can be found in[22]-[25].…”

A fresh view of cosmological models describing very early universe: General solution of the dynamical equations

“…The cosmological models are derived by a further dimensional reduction of the two-dimensional scalaron or vecton theories. With this aim, we apply a kind of a direct separating the r and t variables that does not require any group-theoretical considerations (see, e.g., discussion in [4]- [5]). Thus we get nonlinear dynamical systems of functions that depend on t or on r and can describe cosmological models or static states.…”

“…A detailed description of the reduction procedure was given in our earlier work, e.g.,[4],[5] 4. We call the 'special' anisotropic the cosmology with β ′ = γ ′ andα = 0, which is dual to FRLW.…”