### Abstract or Additional Information

**Abstract**: The interplay between dispersion and nonlinearity in many physical systems leads to the formation of solitary waves, localized coherent structures that carry energy through the system. For example, such waves were experimentally observed in granular materials and are believed to be responsible for energy transport in muscle proteins. We consider solitary waves in a one-dimensional lattice with nonlinear first-neighbor and harmonic second-neighbor interactions. We show that when the interactions are strongly competitive, such waves must be *strictly supersonic*, in the sense that solitary wave solutions do not exist in a velocity gap above the sonic limit and below a certain minimal velocity threshold. Solutions with velocities above the threshold bifurcate from a *short-length* linear wave and have damped oscillatory tails, thus representing a discrete analog of capillary-gravity waves. We derive modulation equations allowing us to construct these solutions to the leading order near the bifurcation point and use this approximation to obtain the solitary waves numerically at larger velocities. If time permits, quasicontinuum descriptions and construction of exact solutions of this type for a lattice with piecewise linear nearest-neighbor interactions will also be discussed. This talk is based on joint work with Lev Truskinovsky.