Optical quantum recursive vortex filament flows and energy with the bishop frame

Abstract

In this study, we obtain the energy equations for recursive and non-recursive vortex filament flows with bishop (b) frame in Euclidean 3-space. By doing this, we make a specific contribution to geometrical expression of energy of a particle moving in space. Here, we first formulate the vector field energy on any surface formed by a moving curve with arc-length variable. Next, we calculate energy formulas for a vortex filament flow and for a flow making recursively vortex filament. For this, we use the recursion operator and the Da Rios (vortex filament) equation. The Da Rios equation is one of important examples of nonlinear equation systems and have served to topics such as aerodynamic systems, fluid dynamics and magnetic vortex filament flows. Finally, we give some physical and geometric simulations with examples of recursive and non-recursive vortex filament flow.