Kazan, AhmetAltin, Mustafa2026-06-192026-06-1920250034-4877https://doi.org/10.1016/S0034-4877(25)00011-4https://hdl.handle.net/20.500.12899/4865In this paper, for a soliton surface Ω = Ω(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 Ω = Ω(u, v) for all v. Also, we get the geometric invariants k and h of the soliton surface Ω = Ω(u, v) and we obtain the Gaussian curvature, mean curvature vector and Gaussian torsion of Ω. 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. © 2025 Polish Scientific Publisherseninfo:eu-repo/semantics/openAccessBetchov–Da Rios EquationCurvature EllipseExtended Darboux FrameWintgen InequalityArticle10.1016/S0034-4877(25)00011-495171922-s2.0-105000606502Q2