Dynamics of χ$\text{}^{(2)}$ Solitary Waves: a Multiple Scales Approach

The dynamics of solitary waves in second-order nonlinear materials are discussed using a multiple scales model. After making some comments on the applicability of other perturbation techniques the multiple scales approach is developed with a view to setting up a line of approach that, in principle, permits radiative effects to be modelled. After a closure condition is applied, equations for the evolution of dynamical variables are developed. Applications of these equations to loss and interactions are presented together with confirmation from numerical simulations. It is emphasised that the method is capable of extension to higher-order perturbations and, hence, into the solitary wave fusion region. The established interpretation of quasi-phase-matching fluctuations as loss is discussed and the simple problems of soliton (solitary wave) pair interactions in both loss-free and lossy media are analysed.