Symmetries shape the current in ratchets induced by a biharmonic driving force

Equations describing the evolution of particles, solitons, or localized structures, driven by a zero-average, periodic, external force, and invariant under time reversal and a half-period time shift, exhibit a ratchet current when the driving force breaks these symmetries. The biharmonic force f(t)=ϵ₁ cos(qωt+φ₁)+ϵ₂ cos(pωt+φ₂) does it for almost any choice of φ₁ and φ₂, provided p and q are two coprime integers such that $p+q$ is odd. It has been widely observed, in experiments in semiconductors, in Josephson junctions, photonic crystals, etc., as well as in simulations, that the ratchet current induced by this force has the shape v∝ϵ₁ᵖϵ₂q cos(pφ₁-qφ₂+θ₀) for small amplitudes, where θ₀ depends on the damping (θ₀=π/2 if there is no damping, and θ₀=0 for overdamped systems). We rigorously prove that this precise shape can be obtained solely from the broken symmetries of the system and is independent of the details of the equation describing the system.