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We study long nonlinear longitudinal bulk strain waves in a hyperelastic rodof circular cross section within the scope of the general weakly-nonlinearelasticity leading to a model with quadratic and cubic nonlinearities. Wesystematically derive the extended Boussinesq and Korteweg - de Vries - typeequations and construct a family of approximate weakly-nonlinear solitonsolutions with the help of near-identity transformations. These solutions arecompared with the results of direct numerical simulations of the originalnonlinear problem formulation, showing excellent agreement within the range oftheir asymptotic validity (waves of small amplitude) and extending theirrelevance beyond it (to the waves of moderate amplitude) as a very good initialguess. In particular, we were able to observe a stably propagating "table-top"soliton..

Longitudinal bulk strain solitons in a hyperelastic rod with quadratic and cubic nonlinearities