## Abstract

We introduce a model of a lossy second-harmonic-generating [Formula presented] cavity externally pumped at the third harmonic, which gives rise to driving terms of a new type, corresponding to a cross-parametric gain. The equation for the fundamental-frequency (FF) wave may also contain a quadratic self-driving term, which is generated by the cubic nonlinearity of the medium. Unlike previously studied phase-matched models of [Formula presented] cavities driven at the second harmonic or at FF, the present one admits an exact analytical solution for the soliton, at a special value of the gain parameter. Two families of solitons are found in a numerical form, and their stability area is identified through numerical computation of the perturbation eigenvalues (stability of the zero solution, which is a necessary condition for the soliton’s stability, is investigated in an analytical form). One family is a continuation of the special analytical solution. At given values of the parameters, one soliton is stable and the other one is not; they swap their stability at a critical value of the mismatch parameter. The stability of the solitons is also verified in direct simulations, which demonstrate that an unstable pulse rearranges itself into a stable one, or into a delocalized state, or decays to zero. A soliton which was given an initial boost [Formula presented] starts to move but quickly comes to a halt, if the boost is smaller than a critical value [Formula presented]. If [Formula presented], the boost destroys the soliton (sometimes, through splitting into two secondary pulses). Interactions between initially separated solitons are investigated, too. It is concluded that stable solitons always merge into a single one. In the system with weak loss, it appears in a vibrating form, slowly relaxing to the static shape. With stronger loss, the final soliton emerges in the stationary form.

Original language | English |
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Pages (from-to) | 10 |

Number of pages | 1 |

Journal | Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics |

Volume | 70 |

Issue number | 6 |

DOIs | |

State | Published - 2004 |