Study of transients in the propagation of nonlinear waves in some reaction diffusion systems

We study the transient dynamics of single species reaction diffusion systems whose reaction terms $f(u)$ vary nonlinearly near $u\approx 0$, specifically as $f(u)\approx u^2$ and $f(u)\approx u^3$. We consider three cases, calculate their traveling wave fronts and speeds \emph{analytically} and solve the equations numerically with different initial conditions to study the approach to the asymptotic front shape and speed. Observed time evolution is found to be quite sensitive to initial conditions and to display in some cases nonmonotonic behavior. Our analysis is centered on cases with $f^{\mathrm{\prime }}(0)=0$, and uncovers findings qualitatively as well quantitatively different from the more familiar reaction diffusion equations with $f^{\mathrm{\prime }}(0)>0$. These differences are ascribable to the disparity in time scales between the evolution of the front interior and the front tail.