A Geometric Application of Soliton Surfaces associated with the Betchov-Da Rios Equation using an Extended Darboux Frame Field in $E^4$

In this paper, for a soliton surface $\mathrm{\Omega }=\mathrm{\Omega }(u,v)$ associated with the Betchov-Da Rios equation, we obtain the derivative formulas of an extended Darboux frame field of a unit speed curve $u$-parameter curve $\mathrm{\Omega }=\mathrm{\Omega }(u,v)$ for all $v$. Also, we get the geometric invariants $k$ and $h$ of the soliton surface $\mathrm{\Omega }=\mathrm{\Omega }(u,v)$ and we obtain the Gaussian curvature, mean curvature vector and Gaussian torsion of $\mathrm{\Omega }$. We give some important geometric characterizations such as flatness, minimality and semi-umbilicaly with the aid of these invariants. Additionally, we study the curvature ellipse of the Betchov-Da Rios soliton surface and Wintgen ideal (superconformal) Betchov-Da Rios soliton surface with respect to an extended Darboux frame field. Finally, we construct an application for the Betchov-Da Rios soliton surface with the aid of an extended Darboux frame field.