Bäcklund transformations as nonlinear ordinary differential, or difference equations with superposition formulas (Q2781710)

The author briefly reviews the theory of ordinary differential equations with a nonlinear superposition principle, i.e. equations where the general solution can be expressed as a function of finitely many particular solutions. It is shown that such an equation is associated with a pair of finite-dimensional Lie algebras \((L_0,L)\) with \(L_0\subset L\). The article concentrates on the case that \(L\) is a simple Lie algebra and \(L_0\) a maximal parabolic subalgebra. The superposition formula arises from the corresponding group action.NEWLINENEWLINENEWLINEFinally, the problem of discretizing such an equation with a superposition principle is considered. It is shown that each of these differential equations leads naturally to a difference equation by simply replacing the algebra action by the corresponding group action. This difference equation has the same solution as the original differential equation.NEWLINENEWLINEFor the entire collection see [Zbl 0974.00040].