Ferromagnet-Type System: Integrable Flows of Curves/Surfaces, Soliton Solutions, and Equivalence
Nugmanova G. Bekova G. Zhassybayeva M. Taishiyeva A. Yesmakhanova K. Myrzakulova Z.
July 2025Multidisciplinary Digital Publishing Institute (MDPI)
Symmetry
2025#17Issue 7
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.
ferromagnet-type system , fractional CCD equation , gauge equivalent , integrable flows of curves/surfaces , soliton solution
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Department of Mathematical and Computer Modeling, L.N. Gumilyov Eurasian National University, Astana, 010000, Kazakhstan
Department of General and Theoretical Physics, L.N. Gumilyov Eurasian National University, Astana, 010000, Kazakhstan
Department of Mathematics and Methods of Teaching Mathematics, Kh. Dosmukhamedov Atyrau University, Atyrau, 060000, Kazakhstan
Department of Algebra and Geometry, L.N. Gumilyov Eurasian National University, Astana, 010000, Kazakhstan
Department of Mathematical and Computer Modeling
Department of General and Theoretical Physics
Department of Mathematics and Methods of Teaching Mathematics
Department of Algebra and Geometry
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