Motions of Curves in the Projective Plane Inducing the Kaup-Kupershmidt Hierarchy

TL;DR

This paper derives equations for curve motions in the projective plane that generate the Kaup-Kupershmidt hierarchy, introducing new local flows and analyzing their solutions, including traveling wave cnoidal solutions.

Contribution

It constructs a family of local flows inducing the Kaup-Kupershmidt hierarchy and discusses the integration of congruence curves and specific solutions.

Findings

Derived equations for curve motions in the projective plane.

Constructed local flows inducing the Kaup-Kupershmidt hierarchy.

Described traveling wave cnoidal solutions of the hierarchy.

Abstract

The equation of a motion of curves in the projective plane is deduced. Local flows are defined in terms of polynomial differential functions. A family of local flows inducing the Kaup-Kupershmidt hierarchy is constructed. The integration of the congruence curves is discussed. Local motions defined by the traveling wave cnoidal solutions of the fifth-order Kaup-Kupershmidt equation are described.