One-dimensional monatomic lattices with Hamiltonian H = n∈ Z Z (1 2 p 2 n + V (q n+1 − q n)) are known to carry localized travelling wave solutions, for generic nonlinear potentials V [FW94]. In this paper we derive the asymptotic profile of these waves in the high-energy limit H → ∞, for Lennard-Jones type interactions. The limit profile is proved to be a universal, highly discrete, piecewise linear wave concentrated on a single atomic spacing. This shows that dispersionless energy transport in these systems is not confined to the long-wave regime on which the theoretical literature has hitherto focused, but also occurs at atomic-scale localization.

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Abstract. It is well established that the solitary waves of FPU-type chains converge in the high-energy limit to traveling waves of the hard-sphere model. In this paper we establish improved asymptotic expressions for the wave profiles as well as an explicit formula for the wave speed. The key step in our approach is the derivation of an asymptotic ODE for the appropriately rescaled strain profile.

A system of ordinary differential equations describing the interaction of a fast and a slow particle is studied, where the interaction potential Uε depends on a small parameter ε. The parameter ε can be interpreted as the mass ratio of the two particles. For positive ε, the equations of motion are Hamiltonian. It is known [6] that the homogenised limit ε → 0 results again in a Hamiltonian system with homogenised potential U hom . In this article, we are interested in the situation where ε is small but positive. In the first part of this work, we rigorously derive the second-order correction to the homogenised degrees of freedom, notably for the slow particle yε = y0 + ε 2 (ȳ2 + [y2] ε ). In the second part, we give the resulting asymptotic expansion of the energy associated with the fast particle), a thermodynamic interpretation. In particular, we note that to leading-order ε → 0, the dynamics of the fast particle can be identified as an adiabatic process with constant entropy, dS0 = 0. This limit ε → 0 is characterised by an energy relation that describes equilibrium thermodynamic processes, dE ⊥ 0 = F0dy0 + T0dS0 = F0dy0, where T0 and F0 are the leading-order temperature and external force terms respectively. In contrast, we find that to second-order ε 2 , a non-constant entropy emerges, d S2 = 0, effectively describing a non-adiabatic process. Remarkably, this process satisfies on average (in the weak * limit) a similar thermodynamic energy relation, i.e., d Ē⊥ 2 = F0dȳ2 + T0d S2.