Lie-theoretical generalization and discretization of the Pinney equation
The authors discuss two types of Pinney equations which admit nonlinear superposition principles. An introduction is given to superposition principles based on classical results of Lie for the structure of corresponding differential equations. Then the Pinney equation \[ \ddot u+w(t) \cdot u={\alpha \over u^3} \] is written as a first-order system, and the corresponding Lie algebra \(Sl(2)\) of generators is transformed to a normal form depending on \(\alpha\) and the fields \(\mathbb{R}\) or \(\mathbb{C}\). On this background suitable transformations are constructed, and formulas for the general solution using two solutions and the superposition principle. Connections to nonlinear Schrödinger equations and discretization formulas are given.
scientific article; zbMATH DE number 1115533
scientific article