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A potential dynamics approach is developed to determine the periodic standingand traveling wave patterns associated with self-propelling camphor objectsfloating on ring-shaped water channels. Exact solutions of the wave patternsare derived. The bifurcation diagram describing the transition between theimmobile and self-propelling modes of camphor objects is derivedsemi-analytically. The bifurcation is of a pitchfork type which is consistentwith earlier theoretical work in which natural boundary conditions have beenconsidered.

Exact solutions for chemical concentration waves of self-propelling camphor particles racing on a ring: A novel potential dynamics perspective