  We study long nonlinear longitudinal bulk strain waves in a hyperelastic rod
of circular cross section within the scope of the general weakly-nonlinear
elasticity leading to a model with quadratic and cubic nonlinearities. We
systematically derive the extended Boussinesq and Korteweg - de Vries - type
equations and construct a family of approximate weakly-nonlinear soliton
solutions with the help of near-identity transformations. These solutions are
compared with the results of direct numerical simulations of the original
nonlinear problem formulation, showing excellent agreement within the range of
their asymptotic validity (waves of small amplitude) and extending their
relevance beyond it (to the waves of moderate amplitude) as a very good initial
guess. In particular, we were able to observe a stably propagating "table-top"
soliton..

Pattern Formation and Solitons;Classical Physics