A fifth-order dispersive partial differential equation for curve flow on the sphere

The initial value problem for a fifth-order nonlinear dispersive partial differential equation describing the curve flow on the sphere is considered. A typical example of the equation arises in a hierarchy of completely integrable systems containing one-dimensional classical Heisenberg ferromagnetic spin model. This paper establishes the local existence and uniqueness of a solution to the initial value problem under the periodic boundary condition. The proof is based on the energy method combined with a kind of gauge transformation to overcome the difficulty of a loss of derivative.