Asymptotic analysis of weakly nonlinear Bessel–Gauß beams

In this paper we investigate the propagation of conical waves in nonlinear media. In particular, we are interested in the effects resulting from applying a Gaussian apodization to an ideal nondiffracting wave. First, we present a multiple scales approach to derive amplitude equations for weakly nonlinear conical waves from a governing equation of cubic nonlinear Schrِdinger type. From these equations we obtain asymptotic solutions for the linear and the weakly nonlinear problem for which we state several uniform estimates that describe the deviation from the ideal nondiffracting solution. Moreover, we show numerical simulations based on an implementation of our amplitude equations to support and illustrate our analytical results.