On a three-dimensional Riccati differential equation and its symmetries

Abstract

A three-dimensional Riccati differential equation of complex quaternionvalued 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 Schrödinger equation. Using symmetry reductions and relations between the three-dimensional Riccati and the Schrödinger equation, examples are given to obtain solutions of both equations.

Review of Quaternion Differential Equations: Historical Development, Applications, and Future Direction

“…Campos and Waphin [37] established the existence of periodic solutions with quaternion differential equations, and Wilczynski [38] provided some sufficient conditions to periodically solve the Riccati quaternion differential equations. The three-dimensional Riccati quaternion differential equation was carried out by Papillon and Tremblay [39], while Grigorian (2019) showed the general solution criteria. The basic theory of linear quaternion differential equations was provided by Kou and Xia [42] in their study on linear quaternion differential equations.…”

“…Furthermore, Campos and Mawhim demonstrated the existence of periodic solutions to quaternion differential equations [37], while Wilczynski provided some sufficient conditions for periodic solutions of the Riccati quaternion differential equations [38]. Riccati's three-dimensional quaternion differential equations were performed by [39], while Grigorian (2019) found the general solution criteria for the equations [40]. The basic linear quaternion differential equation theory and its application to quantum mechanics, fluid mechanics, Frenet frames in differential geometry, dynamic modeling [41], Kalman filters, etc., are systematically presented by [42].…”