Generalized Kaup?Kupershmidt solitons and other solitary wave solutions of the higher order KdV equations

New types of exact explicit solitary wave solutions of the higher order KdV equations are identified by applying a direct method designed specifically for constructing solitary wave solutions of evolution equations. The first type is the 'generalized Kaup–Kupershmidt' (GKK) solitary waves, which unify the structures of the sech2 KdV-like soliton and the Kaup–Kupershmidt soliton and also provide solutions of some other equations. One of those equations is found to possess the multi-soliton solutions which makes it a good candidate for being integrable in terms of the GKK solitons. Another type of solutions of the higher order KdV equations identified by applying the method represents the steady-state localized structures. The variety of equations possessing such solutions includes the integrable (in terms of the sech2 solitons) Sawada–Kotera equation which thus appears to provide the localized solutions of two types.