Higher-order smooth profiles and breather-positon phenomena in the Kuralay equation

In this paper, we investigated higher-order smooth positon and breather-positon solutions of the Kuralay equation. Starting from the associated Lax pair, we constructed an explicit $N$-fold Darboux transformation (DT) in determinant form. By introducing a spectral parameter degeneration procedure combined with higher-order Taylor expansion, we derived smooth higher-order positon solutions from multi-soliton solutions. Furthermore, under nonvanishing boundary conditions, breather-positon solutions were obtained. The dynamical properties of these solutions were analyzed, revealing elastic interaction behavior and nontrivial phase shifts. The results provided a unified framework for constructing degenerate localized wave structures and extended existing studies on the Kuralay equation.