title: Wavy fronts in a hyperbolic FitzHugh-Nagumo system and the effects of cross diffusion.
abstract: We study a hyperbolic version of the FitzHugh-Nagumo (also known as the Bonhoeffer-van der Pol) reaction-diffusion system. To be able to obtain analytical results, we employ a piecewise linear approximation of the nonlinear kinetic term. The hyperbolic version is compared with the standard parabolic FitzHugh-Nagumo system. We completely describe the dynamics of wavefronts and discuss the properties of the speed equation. The nonequilibrium Ising-Bloch bifurcation of traveling fronts is found to occur in the hyperbolic case as well as in the parabolic system. Waves in the hyperbolic case typically propagate with lower speeds, in absolute value, than waves in the parabolic one. We find the interesting feature that the hyperbolic and parabolic front trajectories coincide in the phase plane for the FitzHugh-Nagumo model with a diagonal diffusion matrix, which is the case of self-diffusion, and differ for the system with cross diffusion.
authors.affiliation: Department of Continuum Mechanics, Dorodnicyn Computing Centre, Russian Academy of Sciences, Vavilova 40, 119333 Moscow, Russia.
authors.forename: Evgeny P
authors.initials: EP
authors.lastname: Zemskov
authors.affiliation: Institute of Theoretical and Experimental Biophysics, Russian Academy of Sciences, Institutskaya 3, 142290 Pushchino, Moscow Region, Russia.
authors.forename: Mikhail A
authors.initials: MA
authors.lastname: Tsyganov
authors.affiliation: Department of Chemistry, Southern Methodist University, Dallas, Texas 75275-0314, USA.
authors.forename: Werner
authors.initials: W
authors.lastname: Horsthemke