Classification of systems of nonlinear ordinary differential equations with superposition principles

Abstract

A system of n first-order nonlinear ordinary differential equations ẋ(t)=f(x,t) is said to admit a superposition principle if its general solution can be written as a function of a finite number m of particular solutions and n constants. Such a system can be associated with the nonlinear action of a Lie group G on a space M. We show that ‘‘indecomposable’’ systems of ODE’s with supersposition principles are obtained if and only if the Lie algebras L0⊆L, corresponding to the isotropy group H of a point and the group G, respectively, define a transitive primitive filtered Lie algebra (L,L0). Using known results from the theory of transitive primitive Lie algebras we deduce that L0 must be a maximal subalgebra of L and that G must be an affine group, a simple Lie group, or the direct product of two identical simple Lie groups. Affine groups lead to linear equations, the other types to nonlinear equations with polynomial or rational nonlinearities. Equations corresponding to the classical complex Lie algebras are worked out in detail.