  We study a chain of anharmonic springs with tunable power law interactions as
a minimal model to explore the propagation of strongly non-linear solitary wave
excitations in a background of thermal fluctuations. By treating the solitary
waves as quasi-particles, we derive an effective Langevin equation and obtain
their damping rate and thermal diffusion. These analytical findings compare
favorably against numerical results from a Langevin dynamic simulation. In our
chains composed of two sided non-linear springs, we report the existence of an
expansion solitary wave (anti-soliton) in addition to the compressive solitary
waves observed for non-cohesive macroscopic particles.
