On the FKPP equation with Gaussian shear advection

A boundary value problem for the Fisher–Kolmogorov–Petrovskii–Piskunov (FKPP) equation with shear random advection is investigated. It is demonstrated that the upper bounds for front propagation in the FKPP equation with Gaussian advection obtained by an analysis of its linear ensemble-averaged upper solution can give too rough predictions. It is shown that the unboundedness of the Gaussian advection affects ensemble-averaged solutions of linear and nonlinear problems in a different way. This analytical prediction is confirmed by direct numerical simulations. The phenomenon of propagation failure due to zero boundary conditions is studied and the critical conditions are found. Numerical procedure for the analysis of random wave propagation is suggested by introduction of mean front position and its variance. Some numerical experiments are presented.