On Geometric-Analytic Aspects of Solvable Nonlinear Ordinary Differential Equations and Some Applications

A geometric-analytic approach to studying invariants of solvable nonlinear ordinary differential equations is developed. In particular, there is described in detail a general scheme of constructing solvable nonlinear ordinary differential equations, based on a linear differential spectral problem and its related invariants. Examples of nonlinear differential equations applications are discussed, generalizing those previously studied in the literature. The analytical properties of the invariants and determining the Noether-Lax evolution equation, including its asymptotic properties, are analyzed in detail. Some interesting from a practical point examples of the second ordinary differential equations are analyzed in detail, including the classical Van der Pol and Painlevé equations. The backgrounds of the isolvability problem are also presented and applied to ordinary super-differential equations on the superaxis, which are of interest for research in the field of modern quantum physics.