On Some Properties of Solutions of Nonlinear Systems of Differential Equations

Abstract

The non-autonomous nonlinear system of two first order differential equations with arbitrary parameter and the autonomous system of two nonlinear differential equations with quadratic nonlinearity of derivation of unknown functions, containing arbitrary parameters a, α, β, and γ and nonzero parameters b and c satisfying the conditions (b² – c²)(b² – 4c²)(4b² – c²)  = 0 are studied. The conditions on the parameters of the mentioned systems have been determined under which their general solutions have no moving special critical points, i.e., have the Painlevé property (the systems are Painlevé systems). It is proved that the non-autonomous system for any value of the parameter l is a Painlevé type system and is equivalent in one of its components to the second order differential equation obtained by N.A. Kudryashov. The solution of this equation is expressed in terms of the solution of the second Painlevé equation. Direct and inverse Bäcklund transformations have been constructed for this equation. Each component of the autonomous system is equivalent to two second order nonlinear differential equations. It is examined whether these equations have the Painlevé property depending on parameter values. It is proved that the autonomous system with the parameters b² = c²  ≠ 0 is a Painlevé type system: it is equivalent to second-order differential equations, which are either integrated in elliptic functions or admit linearization. In the other two cases, it has this property if a = 0.