Collapse of a Volterra soliton into a weak monotone shock wave

The perturbation of the stationary solitary solution of a feeder-eater Volterra equation by a small linear dissipative-like term is studied both numerically and analytically and leads to the existence of “quasi-solitons” which are hybrid non-stationary profiles constituted each by a high amplitude, exponentially damped soliton followed by a small amplitude uniform residue left behind the advancing pulse and shown to be a stationary Burgers shock wave. These quasi-solitons appear as stable as unperturbed solitons and preserve their own identity despite nonlinear interactions. They seem to be a consequence of the finiteness of the initial condition norm (measured above the reference noise level).