Symmetries and conservation laws of the generalized Krichever–Novikov equation

Abstract

A computational classification of contact symmetries and higher-order local symmetries that do not commute with , as well as local conserved densities that are not invariant under is carried out for a generalized version of the Krichever–Novikov (KN) equation. Several new results are obtained. First, the KN equation is explicitly shown to have a local conserved density that contains . Second, apart from the dilational point symmetries known for special cases of the KN equation and its generalized version, no other local symmetries with low differential order are found to contain . Third, the basic Hamiltonian structure of the KN equation is used to map the local conserved density containing into a nonlocal symmetry that contains . Fourth, a recursion operator is applied to this nonlocal symmetry to produce a hierarchy of nonlocal symmetries that have explicit dependence on . When the inverse of the Hamiltonian map is applied to this hierarchy, only trivial conserved densities are obtained.