Fifth-order evolution equations describing pseudospherical surfaces

We consider differential equations which describe pseudospherical surfaces, with associated 1-forms , 1⩽i⩽3. We characterize all such equations of type ut=uxxxxx+G(u,ux,…,uxxxx) whose associated 1-forms satisfy fp1=μpf11+ηp, μp,ηp∈R, 2⩽p⩽3, in addition to a generic technical assumption. We also classify all of these equations which are independent of any of the real parameters μp, ηp, obtaining as particular cases the fifth-order Korteweg–de Vries, the Sawada–Kotera and the Kaup–Kupershmidt equations. We determine huge classes of equations describing pseudospherical surfaces and their respective linear problems in which particular cases are obtained by merely specifying certain functions which depend on u and its derivatives with respect to x.