It is shown that, when studying nonlinear longitudinal deformation waves in cylindrical shells, it is possible to obtain physically admissible solitary wave solutions using refined shell models. In the article, a physically admissible exact localized solution based on the Flügge – Lurie – Byrne model is constructed. An analysis of the influence of the external nonlinear elastic medium on the exact solutions obtained is carried out. It is established that the use of quadratic and cubic nonlinear deformation laws leads to nonintegrable equations with exact soliton-like solutions. However, the amplitudes of the exact solutions exceed the values of permissible displacements corresponding to the maximum points on the curves of the deformation laws of the external medium, which leads to the physical inadmissibility of these solutions.

The FitzHugh – Rinzel model is considered, which differs from the famous FitzHugh – Nagumo model by the presence of an additional superslow dependent variable. Analytical properties of this model are studied. The original system of equations is transformed into a third-order nonlinear ordinary differential equation. It is shown that, in the general case, the equation does not pass the Painlevé test, and the general solution cannot be represented by Laurent series. Using the singular manifold method in terms of the Schwarzian derivative, an exact particular solution in the form of a kink is constructed, and restrictions on the coefficients of the equation necessary for the existence of such a solution are revealed. An asymptotic solution is obtained that shows good agreement with the numerical one. This solution can be used to verify the results in a numerical study of the FitzHugh – Rinzel model.