Efficient numerical continuation and stability analysis of spatiotemporal quadratic optical solitons

A numerical method is set out which efficiently computes stationary (z-independent) two- and three-dimensional spatiotemporal solitons in second-harmonic-generating media. The method relies on a Chebyshev decomposition with an infinite mapping, bunching the collocation points near the soliton core. Known results for the type-I interaction are extended and a stability boundary is found by two- parameter continuation as defined by the Vakhitov-Kolokolov criteria. The validity of this criterion is demonstrated in (2+1) dimensions by simulation and direct calculation of the linear spectrum. The method has wider applicability for general soliton-bearing equations in (2+1) and (3+1) dimensions.