Abstract:
This paper investigates an integrable spin system known as the Myrzakulov-XIII (M-XIII)
equation through geometric and gauge-theoretic methods. The M-XIII equation, which
describes dispersionless dynamics with curvature-induced interactions, is shown to admit
a geometric interpretation via curve flows in three-dimensional space. We establish its
gauge equivalence with the complex coupled dispersionless (CCD) system and construct
the corresponding Lax pair. Using the Sym–Tafel formula, we derive exact soliton surfaces
associated with the integrable evolution of curves and surfaces. A key focus is placed
on the role of geometric and gauge symmetry in the integrability structure and solution
construction. The main contributions of this work include: (i) a commutative diagram illustrating the connections between the M-XIII, CCD, and surface deformation models; (ii) the
derivation of new exact solutions for a fractional extension of the M-XIII equation using
the Kudryashov method; and (iii) the classification of these solutions into trigonometric,
hyperbolic, and exponential types. These findings deepen the interplay between symmetry,
geometry, and soliton theory in nonlinear spin systems.
