New analytical solutions for the inextensible Heisenberg ferromagnetic flow and solitonic magnetic flux surfaces in the binormal direction

Maxwellian electromagnetism describes the wave features of the light and related subjects. Its original formulation was established 150 years ago. One of the four Maxwell's equations is Gauss's law, which states significant facts regarding magnetic flux through surfaces. It was also observed that optical media provides surface electromagnetism around 60 years ago. This observation leads to improve new techniques on nano-photonics, metamaterials, and plasmonics. The goal of this manuscript is to suggest novel accurate and local conditions for defining magnetic flux surfaces for the inextensible Heisenberg ferromagnetic flow in the binormal direction. The theoretical accuracy of the methodology is verified through the evolution of magnetic vector fields and the anti-symmetric Lorentz force field operator. On the other hand, the numerical accuracy and efficiency is developed in detail by considering the conformable fractional derivative method when these fields are transformed under the traveling wave hypothesis. (paper)