Nikolay Kudryashov

The integrability of the FitzHugh – Rinzel model is considered. This model is an example of the system of equations having the expansion of the general solution in the Puiseux series with three arbitrary constants. It is shown that the FitzHugh – Rinzel model is not integrable in the general case, but there are two formal first integrals of the system of equations for its description. Exact solutions of the FitzHugh – Rinzel system of equations are given.

A quasi-one-dimensional model of flow of a liquid in a viscoelastic tube is considered. A closed system of the nonlinear equations for the description of perturbations of pressure and radius is propose at flow of a liquid in a is viscoelastic tube. For the analysis of system technique of the multiscale method and the perturbation theory is used. The mathematical model was investigated in case of the large Reynolds numbers. In the equation of movement of a wall of a tube the cubic correction to Hooke’s law is considered. Families of the nonlinear evolutionary equations for the description of perturbations of the basic characteristics of flow are obtained. Exact solutions of some nonlinear evolution equations are found.