Geometric Analysis of Soliton Surfaces via Generalized Bishop Frame Field of Type-C Associated with Betchov–Da Rios Equation in \ (E^{4}\)

In this study, using the generalized Bishop frame field of type-C in four-dimensional Euclidean space, we investigate the geometric properties of a soliton surface M=M(x,y) associated with the Betchov–Da Rios equation. Using a unit speed x-parameter curve M=M(x,y), for all y, we derive the derivative formulas for the generalized Bishop frame field of type-C. We obtain two fundamental geometric invariants of the soliton surface, k and h, characterizing the points of the surface, as well as a few other significant invariants, including Gaussian curvature, a mean curvature vector and Gaussian torsion. We use these surface invariants to prove a collection of theorems that describe the circumstances in which the soliton surface is flat, minimal, semi-umbilic or Wintgen ideal (superconformal). Additionally, we provide a theorem that describes the B-DR soliton surface’s curvature ellipse in relation to the generalized Bishop frame field of type-C in E4. Finally, we construct a foundational example to support our theoretical results and demonstrate the construction of the generalized Bishop frame field of type-C.