  In this paper, we focus on annihilation dynamics for the head-on collision of
traveling patterns. A representative and well-known example of annihilation is
the one observed for 1-dimensional traveling pulses of the FitzHugh-Nagumo
equations. In this paper, we present a new and completely different type of
annihilation arising in a class of three-component reaction diffusion system.
It is even counterintuitive in the sense that the two traveling spots or pulses
come together very slowly but do not merge, keeping some separation, and then
they start to repel each other for a certain time. Finally, up and down
oscillatory instability emerges and grows enough for patterns to become extinct
eventually (see Figs. 1-3). There is a kind of hidden instability embedded in
the traveling patterns, which causes the above annihilation dynamics. The
hidden instability here turns out to be a codimension 2 singularity consisting
of drift and Hopf (DH) instabilities, and there is a parameter regime emanating
from the codimension 2 point in which a new type of annihilation is observed.
The above scenario can be proved analytically up to the onset of annihilation
by reducing it to a finite-dimensional system. Transition from preservation to
annihilation is also discussed in this framework.
