Modulating self-similar waves in a tapered graded-index waveguide

We note that there is a collapse of the self-similar waves recently reported, in a tapered graded-index waveguide. By adding a suitable homogeneous background in the expression for widths of the self-similar solutions which is achievable through the modulation of tapering profile, we could avert this situation, and propagate the self-similar waves quite a distance stably.