TY - JOUR
T1 - Arbitrarily weak head-on collision can induce annihilation
T2 - the role of hidden instabilities
AU - Nishiura, Yasumasa
AU - Teramoto, Takashi
AU - Ueda, Kei Ichi
N1 - Publisher Copyright:
© 2023, The Author(s).
PY - 2023/9
Y1 - 2023/9
N2 - 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 one-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, 2, 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.
AB - 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 one-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, 2, 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.
KW - Center manifold theory
KW - Codimension 2 singularity of drift-Hopf type
KW - Heteroclinic orbit
KW - Hopf–Hopf bifurcation
KW - Pulse interaction
KW - Reaction diffusion system
UR - http://www.scopus.com/inward/record.url?scp=85168388044&partnerID=8YFLogxK
U2 - 10.1007/s13160-023-00607-5
DO - 10.1007/s13160-023-00607-5
M3 - 学術論文
AN - SCOPUS:85168388044
SN - 0916-7005
VL - 40
SP - 1695
EP - 1743
JO - Japan Journal of Industrial and Applied Mathematics
JF - Japan Journal of Industrial and Applied Mathematics
IS - 3
ER -