  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.
