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 TMF, 2010, Volume 162, Number 1, Pages 41–68 (Mi tmf6454)

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

V. de Alfaroa, A. T. Filippovb

a Dipartimento di Fisica Teorica, INFN, Accademia Scienze, Torino, Italy
b Joint Institute for Nuclear Research, Dubna, Moscow Oblast, Russia

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.

Keywords: dilaton gravity, integrable model, Toda equation, Liouville equation

DOI: https://doi.org/10.4213/tmf6454

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English version:
Theoretical and Mathematical Physics, 2010, 162:1, 34–56

Bibliographic databases:

Citation: V. de Alfaro, A. T. Filippov, “Multiexponential models of $(1+1)$-dimensional dilaton gravity and Toda–Liouville integrable models”, TMF, 162:1 (2010), 41–68; Theoret. and Math. Phys., 162:1 (2010), 34–56

Citation in format AMSBIB
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• https://doi.org/10.4213/tmf6454
• http://mi.mathnet.ru/eng/tmf/v162/i1/p41

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. T. Filippov, “Weyl–Eddington–Einstein affine gravity in the context of modern cosmology”, Theoret. and Math. Phys., 163:3 (2010), 753–767
2. Fateev V., Ribault S., “Conformal Toda theory with a boundary”, J. High Energy Phys., 2010, no. 12, 089
3. Proc. Steklov Inst. Math., 272 (2011), 107–118
4. A. T. Filippov, “Unified description of cosmological and static solutions in affine generalized theories of gravity: Vecton–scalaron duality and its applications”, Theoret. and Math. Phys., 177:2 (2013), 1555–1577
5. Davydov E., Filippov A.T., “Dilaton-Scalar Models in the Context of Generalized Affine Gravity Theories: their Properties and Integrability”, Gravit. Cosmol., 19:4 (2013), 209–218
6. A. T. Filippov, “Solving dynamical equations in general homogeneous isotropic cosmologies with a scalaron”, Theoret. and Math. Phys., 188:1 (2016), 1069–1098
7. Filippov T., “A Fresh View of Cosmological Models Describing Very Early Universe: General Solution of the Dynamical Equations”, Phys. Part. Nuclei Lett., 14:2 (2017), 298–303
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