Dynamics of interacting cavity solitons

We derive the equations governing the motion of Kerr solitons in pair waveforms. Recent experiments in microresonators have studied a variety of interaction effects in multisoliton waveforms, including collisions and formation of soliton molecules and crystals. Here we analyze the effective interaction that arises from the coupling of soliton-tail overlap nonlinearity with global soliton variables associated with the breaking of translation symmetry. The interaction is either purely repulsive, or alternates between attraction and repulsion, according to whether the decay of soliton tails is monotone or oscillatory. In the latter case, stable fixed points of the effective dynamical system signify stable soliton molecule configuration, but the exponential weakening of the interaction with increasing inter-soliton separation may prevent the molecule from forming in experimentally accessible time scales. Our theory becomes asymptotically exact in the large-separation limit, and we verify the theoretical calculations using soliton trajectories extracted from direct numerical solutions of the wave equation.