Soliton surfaces by Landau–Lifshitz equations with null cartan curve

This study progresses to a detailed analysis of three-dimensional vector fields, focusing on the differential geometric properties of vector lines, including curvature and torsion, using anholonomic coordinates. Further, it investigates the curl of the binormal vector field in specific configurations, leading to the identification of surfaces with both t congruence and b congruence, supported by the Gauss–Mainardi–Codazzi equations under zero abnormality conditions. This paper also examines the application of the Landau–Lifshitz equation to null Cartan curves, deriving the evolution equation for the Spin vector and exploring its implications for moving null Cartan curves. This study attempts to increase understanding of vector field geometry and its various applications by combining theoretical insights with practical techniques.