Deceleration and Self-Compression of a Wave Pulse in a Discrete Medium

We consider the features of the dynamics of the wave-packet self-action within the framework of a model described by the one-dimensional discrete nonlinear Schrödinger equation with allowance for the effects of acoustic relaxation of the nonlinear response of the medium. Such models are actively used to describe the energy transfer along protein molecules. Analytical and numerical studies show that the dynamics of the wave packets with energies exceeding the critical values significantly differs from the field evolution in a continuous medium. The behavior of initially smooth (on the scale of the distance between the structural elements of the medium) and initially localized field distributions propagating at subsonic speed is studied in detail. A specific self-action regime, which is not characteristic of the continuous limit, is shown to exist where the wave packet, during its propagation, slows down to a complete stop while undergoing self-compression to the size of the lattice period. Radiation losses increase significantly at the final stage of this process and eventually lead to the formation of a soliton-like structure, which usually moves in the opposite direction (with respect to the initial one). In the case of wave packets with supersonic initial speed, the self-action dynamics develops in a similar way. However, in this case, the motion of the quasisoliton which carries most of the energy becomes subsonic during the backward propagation. As applied to the molecular chains, the considered effects lead to a noticeable increase in the localized action of the compressed excitation on individual structural elements of the discrete medium.