Second harmonic generation: Hamiltonian structures and particular solutions

Abstract

For the equations describing second harmonic generation (SHG), we find a linear system different from one found by Kaup (but connected with it by a gauge transformation), which enables us to consider the SHG equations as related to the first negative flow in the coupled KdV hierarchy and to find therefore the Hamiltonian structures and the recursion operator. With the help of the recursion operator we write down the simplest commuting flows and consider their stationary points in order to find particular solutions of the SHG equations. We also investigate some particular invariant solutions of the SHG equations that can be of physical interest, and find their connections with invariant solutions of the Tzitzéica equation. The latter are either connected with Painlevé III or solvable in elliptic functions.