Accommodation of traveling waves of fishers type to the dynamics of the leading tail

Traveling wave solutions for equations of Fisher's type are given by a connection from a saddle point to a node, and their velocity for large times is known to be determined by the spatial decay rate of the initial conditions far ahead of the wave. To understand this phenomenon in simple terms, initial conditions are considered that correspond to a traveling wave, but with an exponential decay rate far ahead of the wave that varies slowly in the region referred to as the leading tail. The wave that results is supersonic in the sense that the speed of the core of the traveling wave is greater than the velocity of the characteristics, or group velocity, in the leading tail. Consequently, the speed of the core adjusts to accommodate the slowly varying decay rate in the leading tail. A generalization is also considered that includes Fisher's equation, for which traveling waves with a stable exponential tail corresponding to a node are supersonic.