The dynamic behavior of a track structure, which is a beam performingbending vibrations and lying on an elastic foundation, is considered. In this case, a generalized foundation model is selected that contains two independent bedding coefficients - rigidity for tensile (compressive) deformation and shear. This model takes into account the distribution capacity of the soil, i.e. its property to settle not only under the loaded area, under the foundation, but also near it. For a beam carrying a moving load, a self-consistent boundary value problem is set, which correctly takes into account the interaction forces in the moving contact. The features of bending wave generation by a zero-frequency oscillation source are studied. Critical velocities of the source are determined. To describe the nonlinear properties of the foundation, the rigidities of elastic elements are considered to depend on the transverse displacement of the beam midline and its gradient. The conditions for the manifestation of modulation instability of weakly nonlinear (quasi-harmonic) waves, leading to their spatial localization and division into separate wave packets, are revealed.