Here we consider dynamical problems as in linear response theory but for purely nonlinear systems where acoustic propagation is prohibited by the potential, e.g., the case of an alignment of elastic grains confined between walls. Our simulations suggest that in the absence of acoustic propagation, the system relaxes using only solitary waves and the eventual state does not resemble an equilibrium state. Further, the studies reveal that multiple perturbations could give rise to hot and cold spots in these systems. We first use particle dynamics based simulations to understand how one of the two unequal colliding solitary waves in the chain can gain energy. Specifically, we find that for head-on collisions the smaller wave gains energy, whereas when a more energetic wave overtakes a less energetic wave, the latter gains energy. The balance between the rate at which the solitary waves break down and the rate at which they grow eventually makes it possible for the system to reach a peculiar equilibriumlike phase that is characteristic of these purely nonlinear systems. The study of the features and the robustness of the fluctuations in time has been addressed next. A particular characteristic of this equilibriumlike or quasiequilibrium phase is that very large energy fluctuations are possible--and by very large, we mean that the energy can vary between zero and several times the average energy per grain. We argue that the magnitude of the fluctuations depend on the nature of the nonlinearity in the potential energy function and the feature that any energy must eventually travel as a compact solitary wave in these systems where the solitary wave energies may vary widely. In closing we address whether these fluctuations are peculiar to one dimension or can exist in higher dimensions. The study hence raises the following intriguing possibility. Are there physical or biological systems where these kinds of nonlinear forces exist, and if so, can such large fluctuations actually be seen? Implications of the study are briefly discussed.

We modify the formulation of Hertz contact theory between two elastic half-solids with axial symmetry and show that these modifications to Hertz's original framework allow the development of force laws of the form F ∝ z n , 1 < n < ∞, where F is the force and z is the distance between the centers of the two solids. The study suggests that it may be possible to design physical systems that can realize such force laws. We let the half-solids be characterized by radii of curvatures R 1 and R 2 and invoke a factor m > 0 to describe any aspect ratio in the two bodies, all being valid near the contact surface. We let the x-y plane be the contact surface with an averaged pressure across the same as opposed to a pressure profile that depends on the contact area of a nonconformal contact as originally used by Hertz. We let the z axis connect the centers of the masses and define z 1,2 = x α /R α−1 1,2 + y α /(mR 1,2 ) α−1 , where z 1,2 0 refers to the compression of bodies 1, 2, α > 1, m > 0, x,y 0. The full cross section can be generated by appropriate reflections using the first quadrant part of the area. We show that the nonlinear repulsive force is F = az n , where n ≡ 1 + 1/α, and z ≡ z 1 + z 2 is the overlap and we present an expression for a = f (E,σ,m,α,R 1 ,R 2 ) with E and σ as Young's modulus and the Poisson ratio, respectively. For α = 2,∞, to similar geometry-dependent constants, we recover Hertz's law and the linear law, describing the repulsion between compressed spheres and disks, respectively. The work provides a connection between the contact geometry and the nonlinear repulsive law via α and m.

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