Research on the Derivation, Numerical Verification of Multitype Soliton Solutions for Integrable Kuralay Equations and Their Implication in Magnetic Spin Dynamics

ABSTRACT This paper takes the integrable Kuralay equations as the research object, aiming to derive various types of soliton solutions and explore the integrable motion of space curves induced by the equations, so as to support the research on nonlinear spin dynamics in the field of magnetic materials. In this paper, the unified ‐expansion method is used to systematically derive soliton solutions expressed by Jacobian elliptic functions. Through parameter degeneration (degenerating into hyperbolic function solutions when the modulus and trigonometric function solutions when ), the evolutionary relationship among different solutions is revealed. Eight types of soliton solutions are obtained in this paper, including periodic trigonometric function solutions, parabolic function solutions, singular solutions, and M‐shaped/W‐shaped solitons (corresponding to Figures 1–8). The parameter configurations and 2D/3D graphical characteristics of each solution are clarified (e.g., kink waves show unidirectional step‐like transitions, and M‐shaped bright waves possess symmetric double peaks). All solitons have clear boundaries without diffusion. For numerical verification, the fourth‐order Runge‐Kutta method combined with Richardson extrapolation is adopted, reducing the calculation error from to . In addition, phase portrait, bifurcation, and initial condition sensitivity analyses are supplemented, and the stability of equilibrium points is classified by the eigenvalues of the Jacobian matrix. In terms of physical implications, the soliton solutions are deeply associated with magnetic spin systems. For instance, kink waves correspond to the migration of spin domain walls, supporting the reading and writing operations of magnetic storage; M‐shaped/W‐shaped solitons contribute to the realization of multistate and high‐density storage. The quantitative influences of parameters on the low‐power consumption and high‐capacity performance of devices are clarified, providing theoretical support and practical guidance for the research on nonlinear spin dynamics and the design of magnetic storage and magneto‐optical modulation devices.