On a three-dimensional Riccati differential equation and its symmetries

TL;DR

This paper investigates a three-dimensional quaternionic Riccati differential equation, exploring its properties, symmetries, and connections to the Schrödinger equation, providing methods to find solutions through symmetry reductions.

Contribution

It introduces the study of quaternionic Riccati equations, derives their symmetries, and links them to the Schrödinger equation for solution generation.

Findings

Derived properties similar to classical Riccati equations

Calculated Lie point symmetries of the quaternionic Riccati equation

Presented examples of solutions via symmetry reductions

Abstract

A three-dimensional Riccati differential equation of complex quaternion-valued functions is studied. Many properties similar to those of the ordinary differential Riccati equation such that linearization and Picard theorem are obtained. Lie point symmetries of the quaternionic Riccati equation are calculated as well as the form of the associated three-dimensional potential of the Schr\"odinger equation. Using symmetry reductions and relations between the three-dimensional Riccati and the Schr\"odinger equation, examples are given to obtain solutions of both equations.