Generalized solutions of the Riccati equation

Abstract

In this paper, we consider a nonlinear differential equation of the first order of the Riccati hierarchy. The concept of a generalized solution for such an equation cannot be introduced within the framework of the classical theory of generalized functions because the product of generalized functions is not defined. To introduce the concept of a generalized solution, two approaches are considered. In the first approach, approximation by solutions of the Cauchy problem with complex initial conditions is used, and generalized solutions are defined as limits of approximating families in the sense of convergence in D′(¡). It is shown that there are two generalized solutions of the Cauchy problem. The type of solution depends on whether the poles of the approximating solution are located in the upper or lower half-plane. The second approach uses approximation with a system of equations. It is shown that there are many approximating systems, meanwhile, generalized solutions of the Cauchy problem depend on the choice of the approximating system.