Solitary waves in a nonintegrable chain with double-well potentials.

Abstract

We study solitary waves in a one-dimensional lattice of identical masses that are connected in series by nonlinear springs. The potential of each spring is nonconvex, where two disjoint convex regions, phase I and phase II, are separated by a concave, spinodal region. Consequently, the force-strain relation of the spring is nonmonotonous, which gives rise to a bistable behavior. Based on analytical treatment, with some approximations, combined with extensive numerical simulations, we are able to reveal important insights. For example, we find that the solitary-wave solution is indifferent to the energy barrier that separates the two energy wells associated with phase I and phase II, and that the shape of the wave can be described by means of merely two scalar properties of the potential of the springs, namely, the ratio of stiffness in phase II and phase I, and the ratio between the Maxwell's force and corresponding transition strain. The latter ratio provides a useful measure for the significance of the spinodal region. Linear stability of the solitary-wave solution is studied analytically using the Vakhitov-Kolokolov criterion applied to the approximate solutions obtained in the first part. These results are validated by numerical simulations. We find that the solitary-wave solution is stable provided that its velocity is higher than some critical value. It is shown that, practically, the solitary waves are stable for almost the entire range of possible wave velocities. This is also manifested in the interaction between two solitary waves or between a solitary wave and a wall (rigid boundary). Such interaction results in a minor change of height and shape of the solitary wave along with the formation of a trail of small undulations that follow the wave, as expected in a nonintegrable system. Even after a significant number of interactions the changes in the wave height and shape are minor, suggesting that the bistable chain may be a useful platform for delivering information over long distances, even concurrently with additional information (other solitary waves) passing through the chain.