Propagation in heterogeneous media

Localized waves are one of the main carriers of information and the effect of heterogeneity of the media in which it propagates is of great importance for the understanding of signaling processes in biological and chemical problems. A typical and simple heterogeneity is a spatially localized bump or dent in 1D or 2D, which in general creates associated defects in the media. One of the main issues is how the geometry of heterogeneity influences over the dynamics of waves. Here the geometry means slope, height, size, curvature and so on. Localized waves are sensitive to those factors and in fact present a variety of dynamics including rebound, pinning, splitting, and traveling motion around the defect. A reduction method to finite-dimensional system is presented, which clarifies the mathematical structure for those dynamics.

In the reference below we mainly focus on a class of one-dimensional traveling pulses the associated parameters of which are close to drift and/or saddle-node bifurcations. The great advantage to study the dynamics in such a class is two-fold: firstly it gives us a perfect microcosm for the variety of outputs in general setting when pulses encounter heterogenieties. Secondly it allows us to reduce the original PDE dynamics to tractable finite dimensional system. Such pulses are sensitive when they run into the heterogeneities and show rich responses such as annihilation, pinning, splitting, rebound as well as penetration. The reduced ODEs explain all these dynamics and the underlying bifurcational structure controlling the transitions among different dynamic regimes. It turns out that there are hidden ordered patterns associated with the critical points of ODEs which play a pivotal role to understand the responses of the pulse. We mainly focus on a bump and periodic types of heterogeneity, however our approach is also applicable to general case. It should be noted that there appears spatio-temporal chaos for periodic type of heterogeneity when its period becomes comparable with the size of the pulse

[1]Y. Nishiura, T. Teramoto, X. Yuan and K. Ueda : "Dynamics of traveling pulses in heterogeneous media", Chaos, 17(3) : 037104 (2007)

[2]X.Yuan, T.Teramoto, and Y.Nishiura:"Heterogeneity-induced defect bifurcation and pulse dynamics for a three-component reaction-diffusion system," Physical Review E, Vol.75(3), 2007: DOI: 10.1103/PhysRevE.75.036220

[3]Yasumasa Nishiura, Yoshihito Oyama, and Kei-ichi Ueda, Dynamics of traveling pulses in heterogeneous media of jump type, Hokkaido Math.J.Vol.36, No.1(2007) pp207-pp242.

[4]Takagi Seiji, Nishiura Yasumasa, Nakagaki Toshiyuki, Ueda Tetsuo and Ueda Kei-ichi,Indecisive behavior of amoeba crossing an environmental barrier, Proceedings of Int. Symp. on Topological Aspects of Critical Systems and Networks, World Scientific Publishing Co. 86-93 (2007). (This is about the response of amoeba when it propagates and encounters with toxic (heterogeneous) area.)