Symmetries and conservation laws of the generalized Krichever-Novikov equation

TL;DR

This paper classifies symmetries and conserved densities of a generalized Krichever-Novikov equation, revealing new local conserved densities involving time and space, and constructing a hierarchy of nonlocal symmetries with explicit t,x dependence.

Contribution

It provides a detailed classification of contact and higher-order symmetries, identifies a new local conserved density involving t,x, and constructs a hierarchy of nonlocal symmetries for the generalized Krichever-Novikov equation.

Findings

Existence of a local conserved density containing t,x.

No low-order local symmetries with t,x dependence besides dilational symmetries.

Hierarchy of nonlocal symmetries with explicit t,x dependence generated by a recursion operator.

Abstract

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