The generalized Schamel equation in nonlinear wave dynamics of cylindrical shells

Abstract

The axisymmetric propagation of longitudinal waves in an integrally stiffened physically and geometrically nonlinear cylindrical shell is studied. The case is considered when the parameters characterizing nonlinearity, dispersion and thickness–radius ratio are the same order. Using the multiscale asymptotic method, the generalized Schamel equation is derived from the equations of motion of the shell. This equation contains an additional term with the fifth derivative with respect to the spatial coordinate which characterizes the high-frequency dispersion. It is shown that the derived equation does not pass the Painleve test and therefore is not integrated by the inverse scattering transform. The geometric series method using Pade approximants is applied to construct exact solitary-wave solution of the equation. It has been established that for the existence of an exact bounded amplitude solution a “soft” type of physical nonlinearity is necessary. The Schamel–Kawahara equation containing the combined nonlinearity is considered; its exact soliton-like solution is given. Numerical simulation confirmed that the initial perturbation in the form of the exact solution generates a persistently propagating solitary wave.