The bi-Hamiltonian structure of some nonlinear fifth- and seventh-order differential equations and recursion formulas for their symmetries and conserved covariants

Using a bi-Hamiltonian formulation we give explicit formulas for the conserved quantities and infinitesimal generators of symmetries for some nonlinear fifth- and seventh-order nonlinear partial differential equations; among them, the Caudrey–Dodd–Gibbon–Sawada–Kotera equation and the Kupershmidt equation. We show that the Lie algebras of the symmetry groups of these equations are of a very special form: Among the C∞ vector fields they are generated from two given commuting vector fields by a recursive application of a single operator. Furthermore, for some higher order equations, those multisoliton solutions, which for ‖t‖→∞ asymptotically decompose into traveling wave solutions, are characterized as eigenvector decompositions of certain operators.