Spike autosolitons in Gray-Scott model

We performed a comprehensive study of spike autosolitons: highly localized solitary states, in the classical reaction-diffusion system — the Gray-Scott model. We developed singular perturbation techniques based on the strong separation of length scales to construct asymptotically the solutions in the form of one-dimensional and higher-dimensional radially-symmetric static autosolitons, and two types of traveling autosolitons. We analyzed the properties of these solutions. Static one-dimensional autosoliton exists in a wide range of excitation levels, while higher-dimensional radially-symmetric static autosolitons exist in much narrower ranges of excitation levels. Ultrafast traveling autosoliton exists in a wide range of excitation levels when the inhibitor is slow and does not diffuse. When the inhibitor diffusion is high but the inhibitor is sufficiently slow, a different type of traveling autosoliton is realized. When the excitation level of the system becomes sufficiently high, the behavior of the tail of the traveling autosoliton changes from decaying to oscillatory leading to self-replication of traveling autosoliton. We also studied asymptotically the