Fifth-order evolution equations describing pseudospherical surfaces

We consider differential equations which describe pseudospherical surfaces, with associated 1-forms omega(i) = f(i1)dx+ f(i2)dt, 1 <= i <= 3. We characterize all such equations of type u(t) = u(xxxxx) + G(u, u(x), ... , u(xxxx)) whose associated 1-forms satisfy f(p1) = mu(p)f(11) + eta(p), mu(p), eta(p) is an element of 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 mu(p), eta(p), 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. (C) 2010 Elsevier Inc. All rights reserved.