On the existence of solitary traveling waves for generalized Hertzian chains

Academic Article

Abstract

  • We consider the question of existence of "bell-shaped" (i.e., nonincreasing for x >0 and nondecreasing 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 (Proc. Am. Math. Soc. 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 (Commun. Math. Phys. 161:391, 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. © Springer Science+Business Media, LLC 2012.
  • Authors

    Digital Object Identifier (doi)

    Author List

  • Stefanov A; Kevrekidis P
  • Start Page

  • 327
  • End Page

  • 349
  • Volume

  • 22
  • Issue

  • 3