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<p>This study investigates the geometric dynamics of electromagnetic wave propagation along optical fibers within a non-Newtonian (multiplicative) differential geometry framework. Modeling the optical fiber as a space curve in multiplicative Euclidean 3-space, we construct a specialized anholonomic coordinate system associated with the multiplicative Frenet frame. Within this setup, we reformulate Maxwell’s equations to derive a novel set of Maxwellian curve evolution equations. We rigorously analyze the polarization evolution of the electric and magnetic fields, establishing explicit relationships between the non-Newtonian Berry phase, Rytov parallel transport, and Fermi-Walker transport laws in both the normal (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\nu \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ν</mi> </math></EquationSource> </InlineEquation>) and binormal (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\beta \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation>) directions. Furthermore, we demonstrate that the derived evolution equations satisfy the Mainardi-Gauss-Codazzi-type compatibility conditions. This geometric formulation reveals a deep intrinsic connection between electromagnetic wave propagation and integrable systems, as exemplified by the emergence of the sine-Gordon equation and the Dini surface geometry in the context of constant negative curvature.</p>

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Integrable Maxwellian Evolution and Geometric Phases in Multiplicative Euclidean Space

  • 期刊论文

  • 发表日期：

    2026年4月29日

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  • Ebru Yanık,
  • Hazal Ceyhan,
  • Zehra Özdemir

摘要

This study investigates the geometric dynamics of electromagnetic wave propagation along optical fibers within a non-Newtonian (multiplicative) differential geometry framework. Modeling the optical fiber as a space curve in multiplicative Euclidean 3-space, we construct a specialized anholonomic coordinate system associated with the multiplicative Frenet frame. Within this setup, we reformulate Maxwell’s equations to derive a novel set of Maxwellian curve evolution equations. We rigorously analyze the polarization evolution of the electric and magnetic fields, establishing explicit relationships between the non-Newtonian Berry phase, Rytov parallel transport, and Fermi-Walker transport laws in both the normal ( \(\nu \) ν ) and binormal ( \(\beta \) β ) directions. Furthermore, we demonstrate that the derived evolution equations satisfy the Mainardi-Gauss-Codazzi-type compatibility conditions. This geometric formulation reveals a deep intrinsic connection between electromagnetic wave propagation and integrable systems, as exemplified by the emergence of the sine-Gordon equation and the Dini surface geometry in the context of constant negative curvature.