Superposition formulae of a fifth order KdV equation and its modified equation (Q915018)

Associated with the Korteweg-de Vries (KdV) equation is a hierarchy of partial differential equations of evolutionary form which describe symmetries of the KdV itself. The next simplest member, to the KdV, belonging to this hierarchy is the so-called fifth-order KdV. Like the KdV itself this equation can be written in bilinear form using the techniques due to \textit{R. Hirota} [Solitons, Top. Curr. Phys. 17, 157-176 (1980) (for the entire collection see Zbl 0428.00010)] provided that, with him, one introduces an auxiliary independent variable. Without this convenience one must deal with a quadrilinear, rather than bilinear, form. The present authors rederive, somewhat labouriously, the known results [ibid.] regarding the Bäcklund transformation for the fifth- order KdV from the quadrilinear form. They summarise the corresponding results for the case of the modified fifth-order KdV.