A nonlinear superposition formula for the Kaup-Kupershmidt partial differential equation

The fifth order Kaup-Kupershmidt (KK) equation is one of the solitonic equations related to the integrable cases of the Henon-Heiles system. Contrary to the Sawada-Kotera equation which is its dual equation, the construction of the N-soliton solutions of KK is not an easy task without using a perturbation scheme. From the auto-Backlund transformation for KK recently obtained with a non zero Backlund parameter, we here derive the nonlinear superposition principle which was previously unknown. This is described by a second-order, second-degree non-linear differential equation of the Appell type linearisable into a third order equation. As a particular solution of this last equation, we derive the expression of the tau-function associated with the 2-soliton solution and by inference we obtain the tau-function for building the N-soliton.