Geodesic
Nonlinear realization of the conformal group in two dimensions and the Liouville equation
Teoretičeskaâ i matematičeskaâ fizika, Tome 58 (1984) no. 2, pp. 200-212 Cet article a éte moissonné depuis la source Math-Net.Ru

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It is shown that the Liouville equation $u_{+-}=m^2e^{-2u}$ has an adequate description in the language of the nonlinear realization of the infinite-parameter conformal group $G$ in two dimensions. The coordinates $x^+$, $x^-$ of the two-dimensional Minkowski space and the field $u(x)$ are identified with certain parameters of the factor space $G/H$, where $H=SO(1,1)$ is the Lorentz group in two dimensions. The Liouville equation arises as one of the covariant conditions of reduction of the factor space $G/H$ to its connected geodesic subspace $SL(2,R)/H$. The alternative reduction to the subspace $\mathscr P(1,1)/H$ where $\mathscr P(1,1)$ is the two-dimensional Poincaré group, leads to the free equation for $u(x)$. The corresponding representations of zero curvature and B cklund transformations acquire in the present approach a simple group-theoretical meaning. The possibility of generalizing the proposed construction to other integrable systems is discussed. 
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     title = {Nonlinear realization of the conformal group in two dimensions and the {Liouville} equation},
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E. A. Ivanov; S. O. Krivonos. Nonlinear realization of the conformal group in two dimensions and the Liouville equation. Teoretičeskaâ i matematičeskaâ fizika, Tome 58 (1984) no. 2, pp. 200-212. http://geodesic.mathdoc.fr/item/TMF_1984_58_2_a4/

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