On the spectral stability of soliton-like solutions to a non-local hydrodynamic-type model

•The conditions for existence of soliton-like traveling wave solutions to a non-local hydrodynamic model are stated.•As a stability test, conditions providing the minimum of Hamiltonian subject to the constant momentum with respect to a uniform stretching of system’s component is considered.•The studies of spectral stability of soliton-like solutions incorporating the consideration of the operator of linearization about the traveling wave, estimations of the maximal number of unstable modes and stating the conditions for their absence are presented. A model of nonlinear elastic medium with internal structure is considered. The medium is assumed to contain cavities, microcracks or blotches of substances that differ sharply in physical properties from the base material. To describe the wave processes in such a medium, the averaged values of physical fields are used. This leads to nonlinear evolutionary PDEs, differing from the classical balance equations. The system under consideration possesses a family of invariant soliton-like solutions. These solutions are shown to be spectrally stable under certain restrictions on the parameters.