Axisymmetric longitudinal waves in a cylindrical shell interacting with a nonlinear elastic medium

Abstract

The evolution of axisymmetric longitudinal waves in a cylindrical Kirchhoff–Love shell interacting with a nonlinear elastic medium is investigated. A generalized model of a nonlinear elastic medium is introduced into consideration, which is reduced in particular cases to the well-known two-parameter Pasternak and Hetenyi models. Using the asymptotic multiscale method for the component of the longitudinal displacement, a non-integrable quasi-hyperbolic equation of the sixth order is derived, generalizing the Ostrovsky equation, and classes of its exact solutions are constructed in terms of the Jacobi elliptic function. It is demonstrated that exact periodic solutions are physically realizable, while solitary wave solutions are nonphysical. The further asymptotic analysis of the derived equation made it possible to obtain the nonlinear Schrödinger equation and use the Lighthill criterion. Relationships are established between the parameters of a nonlinear elastic medium, which are necessary for the development of the modulation instability of small periodic waves.