Staggering dominolike blast front motion in a one-dimensional cold gas

One-dimensional alternating particle systems are widely used to study interconnections between the hydrodynamics of blast waves in a gas-like medium and the Newtonian dynamics of its corpuscular constituents. In this article, we study the model in which point particles with masses m,μ,m,μ,⋯, (m≥μ) are distributed on the positive half-line R_{+}. Their dynamics are initiated by giving a positive velocity to the leftmost particle; in its course, the particles undergo elastic collisions. For this model with m/μ=2, it has been established that the dynamics that start from random initial positions are consistent with predictions based on Euler's hydrodynamic equation. In particular, they have the following properties: (i) the position of the rightmost particle (shock front) evolves as t^{δ} with δ<1; (ii) recoiled particles behind the front enter the negative half-axis; and (iii) particles with locations x≤0 move ballistically and eventually take over the total energy of the system. In this paper, we present numerical and analytical results for the dynamics of this model with nonrandom (typically equidistant) initial positions and various values of m/μ. For m/μ=2 and equidistant initial positions, our results qualitatively agree with those just mentioned. At the same time, we found an infinite family of numbers {M_{k}} such that, for m/μ=M_{k}, the hydrodynamic behavior mentioned changes drastically to the following. At each moment, only a single triplet m,μ,m is in motion, whereas all other particles are at rest. In the moving triplet, two heavy particles move to the right, while the lighter particle oscillates between them, transferring energy and momentum to the right. As a result, the shock front moves ballistically with an average velocity equal to the initial one. Such a 'staggering domino-like' picture is obtained as an exact solution, which yields, in particular, explicit formulas for M_{k} and the particle velocities and positions. Its details and importance in a wider context are also discussed.