Hokkaido Mathematical Journal

NISHIURA Yasumasa, OYAMA Yoshihito, UEDA Kei-ichi,

Dynamics of traveling pulses in heterogeneous media of jump type.

Hokkaido Mathematical Journal, 36 (2007) pp.207-242

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Abstract

We study pulse dynamics in one-dimensional heterogeneous media. In particular we focus on the case where the pulse is close to the singularity of codim 2 type consisting of drift and saddle-node instabilities in a parameter space. We assume that the heterogeneity is of jump type, namely one of the coefficients of the system undergoes an abrupt change at one point in the space. Depending on the height of this jump, the responses of pulse behavior are penetration, splitting, and rebound. Taking advantage of the fact that pulse is close to the singularity, the PDE dynamics can be reduced to a finite-dimensional system, which displays the three behaviors. Moreover it takes a universal form independent of model systems, and is valid for much more general heterogeneities such as bump, periodic, and random cases.

MSC(Primary)34C37
MSC(Secondary)35K57, 35B32
Uncontrolled Keywordsreaction-diffusion equations (35K57), Bifurcation (35B32), Homoclinic and Heteroclinic solutions (34C37)