On a complex differential Riccati equation

TL;DR

This paper investigates a complex nonlinear PDE related to the 2D stationary Schrödinger equation, extending classical Riccati equation properties and introducing new features like an analogue of the Cauchy integral theorem.

Contribution

It generalizes classical Riccati equation results to a complex PDE linked to the 2D Schrödinger equation, including new integral theorem analogues.

Findings

Establishment of properties similar to Euler and Picard theorems for the PDE

Development of an analogue of the Cauchy integral theorem for the equation

Extension of classical Riccati results to a complex PDE context

Abstract

We consider a nonlinear partial differential equation for complex-valued functions which is related to the two-dimensional stationary Schrodinger equation and enjoys many properties similar to those of the ordinary differential Riccati equation as, e.g., the famous Euler theorems, the Picard theorem and others. Besides these generalizations of the classical "one-dimensional" results we discuss new features of the considered equation like, e.g., an analogue of the Cauchy integral theorem.