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On the extended KdV equation and near-identity transformations for strain waves

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HY2W06 - Women in dispersive equations day

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. We systematically derive the extended Boussinesq and Korteweg-de Vries (eKdV) equations and construct a family of approximate weakly-nonlinear soliton solutions with the help of near-identity transformations reducing the eKdV equation to the Gardner equation. 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, e.g. table-top solitons) as a very good initial guess. Recently, the generation of undular bores in polymer bars following tensile fracture was registered using high-speed pointwise photoelasticity. We show that a viscoelastic extended Korteweg – de Vries (veKdV) equation provides a very good agreement with the key observed experimental features for a suitable choice of material parameters.  Based on joint papers with F.E. Garbuzov, Y.M. Beltukov, C.G. Hooper, P.D. Ruiz and J.M. Huntley.

This talk is part of the Isaac Newton Institute Seminar Series series.

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