Kazan, AhmetAltin, Mustafa2026-06-192026-06-1920250034-48771879-0674https://hdl.handle.net/20.500.12899/6228In this paper, for a soliton surface Omega = Omega (u, v) associated with the Betchov-Da Rios equation, we obtain the derivative formulae of an extended Darboux frame field of a unit speed u-parameter curve Omega = Omega(u, v) for all v. Also, we get the geometric invariants k and h of the soliton surface Omega = Omega(u, v) and we obtain the Gaussian curvature, mean curvature vector and Gaussian torsion of 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.eninfo:eu-repo/semantics/closedAccessBetchov-Da Rios EquationExtended Darboux FrameCurvature EllipseWintgen InequalityArticle9517192WOS:001456807200001Q3