Effect of non-local interactions on soliton dynamics in anharmonic chains: Scale competition

We consider the effect of a harmonic non-local interaction potential in a chain with short-range anharmonicity, The existence of two velocity dependent competing length scales leads to two types of solitons with characteristically different widths and shapes for two velocity regions separated by a gap. The low-velocity branch exists up to a maximum critical velocity where the solitary-wave shape reaches a crest-like form. Using direct perturbation methods and a quasicontinuum approximation with the appropriate scale we obtain accurate analytic expressions, Using near threshold stability analysis we find that the crest soliton solution is unstable. In the high-velocity branch we use the multiple scale analysis since two different scales are important in the centre and the tail of the solitary wave, respectively. Here a qualitative agreement is obtained.