Integrable systems as nonlinear realizations of infinite-dimensional symmetries: The Liouville equation example
The Liouville equation \(U_{+-}=m^ 2\exp(-2u)\) is shown to have an adequate description in terms of the nonlinear realization of infinite- parameter conformal group G in \((1+1)\)-dimensions. The \((1+1)\)-Minkowski space coordinates \(x^{\pm},x^ =\) and the field u(x) are identified with certain parameters of the coset G/H, H being the \((1+1)\)-Lorentz group. The Liouville equation appears as one of the covariant constraints reducing this coset space to its connected geodesic subspace \(SL(2,R)/H.\) An alternative reduction to the subspace \({\mathfrak P}(1,1)/H\) (\({\mathfrak P}(1,1)\) is the \((1+1)\)-Poincaré group), yields the free equation for u(x). The corresponding zero-curvature representations naturally arise as the consequences of the Maurer-Cartan equations for G/H and the covariant reduction conditions. Bäcklund transformations also obtain a simple group-theoretical interpretation within this approach: these are represented by the constrained right gauge shifts on the coset G/H. A possibility of extending the method to other integrable system is discussed.
scientific article; zbMATH DE number 3859642