A novel symmetry method for finding exact solutions to nonlinear PDEs is illustrated by applying it to a semilinear reaction-diffusion equation in multi-dimensions. The method uses a separation ansatz to solve an equivalent first-order group foliation system whose independent and dependent variables respectively consist of the invariants and differential invariants of a given one-dimensional gr...

We apply Frobenius integrability theorem in the search of invariants for one-dimensional Hamiltonian systems with a time-dependent potential. We obtain several classes of potential functions for which Frobenius theorem assures the existence of a two-dimensional foliation to which the motion is constrained. In particular, we derive a new infinite class of potentials for which the motion is assur...

On this note we prove that a holomorphic foliation of the projective plane with rich, but finite, automorphism group does not have invariant algebraic curves.

We investigate the vertical foliation of the standard complex contact structure on Γ \ Sl(2,C), where Γ is a discrete subgroup. We find that, if Γ is nonelementary, the vertical leaves on Γ \Sl(2,C) are holomorphic but not regular. However, if Γ is Kleinian, then Γ \ Sl(2,C) contains an open, dense set on which the vertical leaves are regular, complete and biholomorphic to C∗. If Γ is a uniform...