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 Hénon–Heiles system. As opposed to the Sawada–Kotera equation which is its dual equation, the construction of the N-soliton solutions of KK is not an easy task in using a perturbation scheme or the Hirota bilinear formalism. From the Bäcklund transformation obtained by singularity analysis considerations, we here establish the permutability theorem for the KK equation. This allows us to explain the “anomalous” structure of the N-soliton solution, empirically obtained by Parker [Physica D 137 (2000) 25–33, 34–48].