The profile of soliton molecules for integrable complex coupled Kuralay equations

Abstract This study focuses on mathematically exploring the Kuralay equation, which is applicable in diverse fields, such as nonlinear optics, optical fibers, and ferromagnetic materials. This study aims to investigate various soliton solutions and analyze the integrable motion of the induced space curves. This study employs traveling wave transformation, converting the partial differential equation (PDE) into an ordinary differential equation (ODE). Soliton solutions are derived utilizing both the generalized Jacobi elliptic function expansion (JEFE) method and novel extended direct algebraic (EDA) methods. The results encompass a diverse range of soliton solutions, including double periodic waves, shock wave solutions, kink-shaped soliton solutions, solitary waves, bell-shaped solitons, and periodic wave solutions obtained using Mathematica. In contrast, the EDA method produces dark, bright, singular, combined dark-bright solitons, dark-singular combined solitons, solitary wave solutions, etc.. The visual representation of these soliton solutions is accomplished through 3D, 2D, and contour graphics with a meticulous selection of parametric values. The graphical presentation underscores the influence of these parameters on soliton propagation.