The Schamel-Ostrovsky equation in nonlinear wave dynamics of cylindrical shells

On the basis of the smeared stiffener theory, the evolution of nonlinear axisymmetric longitudinal waves in a nonlinear elastic cylindrical shell reinforced by internal stringers is investigated. A non-classical nonlinear physical law is adopted for the shell material, which is characterized by a fractional degree of strain intensity. An analysis of the influence of an external elastic medium on a nonlinear wave process is carried out on the basis of the Winkler and shear-lag model. Using the asymptotic method of multiscale expansions, the Schamel-Ostrovsky quasi-hyperbolic equation for the component of longitudinal displacement was first derived. The Painlevé analysis of the equation showed the impossibility of exact solitary-wave solutions. The dispersion relation is analyzed, the maximum value of the phase velocity of infinitesimal disturbances is determined, above which stationary nonlinear waves can propagate. Based on the finite-difference approach and the Petviashvili method, a numerical simulation of the derived equation is carried out, profiles of stably propagating wave packets and solitary waves are constructed. A special dispersionless case, impossible for an unstiffened shell, has been identified, in which there are exact compactons, peakons and periodic solutions.