Mutual self-trapping of three pulses with multiple current frequencies based on compensation of the induced phase gratings at the middle frequency

Abstract We discuss a mutual self-trapping of the pulses with basic frequency and triple frequency under three-waves interaction in a medium with quadratic nonlinear response. This mode of their interaction is based on a compensation of the induced phase gratings at the middle frequency corresponding to the second harmonic. It leads to absence of the energy exchange between waves with basic and triple frequencies. As a result, the pulses propagate without changing their energies. In the frame-work of long pulse duration approximation we provide a theoretical analysis, from which the conditions on the pulses second-order dispersion are stated and under which the pulses shapes are stable. The stability analysis is provided both theoretically and numerically. We show that weak wave at doubled frequency plays a stabilization role in three-waves interaction under consideration. A presence of weak incident second harmonic wave does not destroy the self-trapping mode of the pulses propagation. The theoretical results are confirmed by computer simulation results which demonstrate the pulses propagation without energy exchange between waves at a distance equal several dispersion lengths.