Moving singularities of the forced Fisher–KPP equation: an asymptotic approach

Abstract

The creation of hairpin or lambda vortices, typical for the early stages of the laminar-turbulent transition process in various boundary layer flows, in some sense may be associated with blow-up solutions of the Fisher-Kolmogorov-Petrovsky-Piskunov equation. In contrast to the usual applications of this nonlinear evolution equation of the reaction-diffusion type, the solution quantity in the present context needs to stay neither bounded nor positive. We focus on the solution behavior beyond a finite-time point blow-up event, which consists of two moving singularities (representing the cores of the vortex legs) propagating in opposite directions, and their initial motion is determined with the method of matched asymptotic expansions. After resolving subtleties concerning the transition between logarithmic and algebraic expansion terms regarding asymptotic layers, we find that the internal singularity structure resembles a combination of second- and first-order poles in the form of a singular traveling wave with a time-dependent speed imprinted through the characteristics of the preceding blow-up event.