Energy transfer between two filaments due to degenerate four-photon parametric process

Recently energy exchange between two filaments crossing at small angle and with power slightly above the critical for self-focusing $P_{cr}$ was experimentally demonstrated. In this paper we present a model describing the process of this transfer through degenerate four-photon parametric mixing. Our model confirms the experimental results that the direction of energy exchange depends on the relative transverse velocity (incident angle), laser intensity and initial distance between the pulses (relative initial phase). We also investigate the interaction between two collinear filaments in order to explain the filaments number reduction for powers close to $P_{cr}$ in multi-filamental propagation.