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

We study long nonlinear longitudinal bulk strain waves in a hyperelastic rod of circular cross section in the framework of general weakly nonlinear elasticity leading to a model with quadratic and cubic nonlinearities. We systematically derive extended equations of the Boussinesq and Korteweg–de Vries types and construct a family of approximate weakly nonlinear soliton solutions using near-identity transformations. We compare these solutions with the results of direct numerical simulations of the original nonlinear problem formulation, showing excellent agreement in the range of their asymptotic validity (waves of small amplitude) and extending their relevance beyond it (to waves of moderate amplitude) as a very good initial condition. In particular, we can observe a stably propagating “table-top” soliton.