On the existence of solitary traveling waves for generalized Hertzian chains

TL;DR

This paper proves the existence of bell-shaped traveling waves in a generalized Hertzian chain model with a specific exponent, using energy minimization techniques, and discusses their decay properties and effects of precompression.

Contribution

It introduces a new variational approach to establish the existence of bell-shaped waves and provides an alternative proof of a known result using this method.

Findings

Existence of bell-shaped traveling waves is established.

Numerical illustrations of decay properties are provided.

Effects of finite precompression on wave decay are discussed.

Abstract

We consider the question of existence of "bell-shaped" (i.e. non-increasing for x>0 and non-decreasing for x<0) traveling waves for the strain variable of the generalized Hertzian model describing, in the special case of a p=3/2 exponent, the dynamics of a granular chain. The proof of existence of such waves is based on the English and Pego [Proceedings of the AMS 133, 1763 (2005)] formulation of the problem. More specifically, we construct an appropriate energy functional, for which we show that the constrained minimization problem over bell-shaped entries has a solution. We also provide an alternative proof of the Friesecke-Wattis result [Comm. Math. Phys 161, 394 (1994)], by using the same approach (but where the minimization is not constrained over bell-shaped curves). We briefly discuss and illustrate numerically the implications on the doubly exponential decay properties of the waves, as well as touch upon the modifications of these properties in the presence of a finite precompression force in the model.