This article is cited in 18 papers

Characteristic Lie rings and integrable models in mathematical physics

A. V. Zhiber^a, R. D. Murtazina^b, I. T. Habibullin^a, A. B. Shabat<sup>c

a</sup> Institute of Mathematics with Computing Centre, Ufa Science Centre, Russian Academy of Sciences, Ufa, Russia
^b Ufa State Aviation Technical University, Ufa, Russia
^c L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences, Moscow, Russia

Abstract: Review is devoted to a systematic exposition of the algebraic approach to the study of nonlinear integrable partial differential equations and their discrete analogues, based on the concept of the characteristic vector field. A special attention is paid to the Darboux integrable equations. The problem of constructing higher symmetries of the equations, as well as their particular and general solutions is discussed. In particular, it is shown that the partial differential equation of hyperbolic type is integrated in quadratures if and only if its characteristic Lie rings in both directions are of finite dimension. For the hyperbolic type equations integrable by the inverse scattering method, the characteristic rings are of minimal growth. The possible applications of the concept of characteristic Lie rings to the systems of differential equations of hyperbolic type with more than two characteristic directions, to the equations of evolution type, and to ordinary differential equations are discussed.

Keywords: characteristic vector field, symmetry, Darboux integrability.

UDC: 517.957

Received: 25.11.2011

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