Spike autosolitons in the Gray-Scott model

We performed a comprehensive study of the spike autosolitons: self-sustained solitary inhomogeneous states, in the classical reaction-diffusion system --- the Gray-Scott model. We developed singular perturbation techniques based on the strong separation of the length scales to construct asymptotically the solutions in the form of a one-dimensional static autosolitons, higher-dimensional radially-symmetric static autosolitons, and two types of traveling autosolitons. We studied the stability of the static autosolitons in one and three dimensions and analyzed the properties of the static and the traveling autosolitons.