Generalized Elastic Line, Differentials of Maps Connected with CNLSE via Conformal Derivative and Magnetic Curves for Darboux Frame

In this paper, we study generalized elastic lines via the intrinsic directional derivatives for the Darboux frame in $\mathbb{R}^{3}$. Subsequently, we investigate the congruences of surfaces associated with the Darboux frame in $\mathbb{R}^{3}$. Furthermore, we establish the Darboux conformal derivative equations and conformal magnetic curves for the Darboux frame in $\mathbb{R}^{3}$. Finally, we present formulas for conformal differentials of maps associated with the CNLSE for the Darboux frame in $\mathbb{R}^{3}$.