On Difference Schemes Approximating First-Order Differential Equations and Defining a Projective Correspondence Between Layers

Abstract

It is well known that there are remarkable differential equations that can be integrated in CAS, but admit several inequivalent approaches to their description. In the present paper, we discuss remarkable differential equations in another sense, namely, equations for which there exist finite difference schemes that preserve the algebraic properties of solutions exactly. It should be noted that this class of differential equations coincides with the class introduced by Painlevé. In terms of the Cauchy problem, a differential equation of this class defines an algebraic correspondence between the initial and final values. For example, the Riccati equation y′ = p(x)y² + q(x)y + r(x) defines a one-to-one correspondence between the initial and final values of y on the projective line. However, the standard finite difference schemes do not preserve this algebraic property of the exact solution. Moreover, the scheme that defines a one-to-one correspondence between the layers actually describes the solution not only before moving singularities but also after them and preserves algebraic properties of equations, such as the anharmonic ratio. After a necessary introduction (Secs. 1 and 2), we describe such a one-to-one scheme for the Riccati equation and prove its properties mentioned above.