Dynamics of pulses and spiral waves in excitable media with an anomalous diffusion

Spiral waves and pulses in the excitable medium with an anomalous diffusion are studied. In the medium with an one-sided fractional diffusion in the x-direction and a normal diffusion in the y-direction, a pulse, traveling along the positive x-direction, has a smaller velocity, which is different from the diffusion of a source in the other media. Its propagating velocity is a linear and increasing function of the square root of diffusion parameter, whose increasing rate depends on the fractional order. A minimal value of the diffusion parameter is needed for successfully propagating pulses, and the threshold becomes large with a decrease of the fractional order. For pulse trains, the frequency-locked bands are shifted along the increasing direction of the perturbation period when the fractional order is decreased. In the propagating process of a spiral wave, the tip drift is induced by the one-sided fractional diffusion, which may be explained by analyzing the SV area in front of the tip.