  A potential dynamics approach is developed to determine the periodic standing
and traveling wave patterns associated with self-propelling camphor objects
floating on ring-shaped water channels. Exact solutions of the wave patterns
are derived. The bifurcation diagram describing the transition between the
immobile and self-propelling modes of camphor objects is derived
semi-analytically. The bifurcation is of a pitchfork type which is consistent
with earlier theoretical work in which natural boundary conditions have been
considered.
