Traveling spike autosolitons in the Gray–Scott model

We developed singular perturbation techniques based on the strong separation of time and length scales to construct the solutions in the form of the traveling spike autosolitons (self-sustained solitary waves) in the Gray–Scott model of an autocatalytic reaction. We found that when the inhibitor diffusion is sufficiently slow, the ultrafast traveling spike autosolitons are realized in a wide range of parameters. When the diffusion of the inhibitor is sufficiently fast, the slower traveling spike autosolitons with the diffusion precursor are realized. We asymptotically calculated the main parameters such as speed and amplitude of these autosolitons as well as the regions of their existence in the Gray–Scott model. We also showed that in certain parameter regions the traveling spike autosolitons coexist with static.