Lienard type equations and their extensions are an important, from the practical viewpoint, class of nonlinear differential equations. These equations appear in the theory of nonlinear oscillations, dynamical systems theory, problems of nonlinear mechanics, elasticity problems, problems of fluid mechanics, problems of description of biological systems dynamics and many other applications. In particular, Lienard type equations include Rayleigh equations used for description of spherical bubble dynamics in liquids, Van der Pol oscillator, Mathews-Lakshmanan oscillator, Van der Pol–Duffing oscillator and series of other equations of nonlinear mechanics.
A project “Nonlocal developments for constructing decisive analytical solutions of nonlinear Lienard type equations and their extensions” is being conducted in a science group of Professor N.A. Kudryashov at the MEPhI Department of Applied mathematics under the supervision of Associate Professor D.I.Sinelschikov as a part of the Russian Federation Presidential Grant for young scientists.
Despite significant practical relevance, the class of Lienard type equations is still understudied from the point of view of construction of obvious analytical solutions. Moreover, detection of equation classes of Lienard type, for which common analytical solution can be obtained in an explicit form, is an important issue. The majority of existing research is based on studying of connection between Lienard type equations and linear differential equations. However, this approach significantly narrows equation classes for which solution construction is possible. That is why it seems to be interesting to research connection between Lienard type equations and integrable nonlinear differential equations (for example, elliptic functions equations).
Thus, the project is aimed at the establishing of connection between Lienard type equations and integrable nonlinear differential equations, construction of point and nonlocal transformations which provide for this connection, construction of common analytical solutions of Lienard type equations, detection of new classes of Lienard type equations.
