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In this manuscript, we try to understand the links between nonlinear equations of motion and equations attained by taking geometric differential description of Hasimoto map of null Cartan curves and some physical results. First, we obtain the equations of Hasimoto transformation occurred by applying Hasimoto map for null Cartan curves. Then, based on the Hasimoto map of null Cartan curves, we examine both Landau–Lifshitz system and visco Landau–Lifshitz system. Lastly, we portray the links of findings obtained by geometric physical issues. After all, we propose a new method, known as Khater method, to obtain solutions of fractional order differential equations. Also, these solutions are simulated by graphically using physical constructions. [ABSTRACT FROM AUTHOR]

In this paper, we construct new efficient insight into the understanding of the configurations of the binormal electromagneto-vortex formation and their consequences in the energy dissipation and magnetic helicity in magnetohydrodynamics (MHD). Firstly, we introduce new formulations of the magnetic helicity of the Lorentz force of the Serret–Frenet frame vectors and associated electromagnetic vector fields of the binormal electromagnetic field lines E − B. Then, we investigate the Godbillon–Vey helicity model. We also compute the time evolution of the binormal electromagnetic field lines of the Frenet–Serret vectors and associated geometric quantities in MHD. Then, we determine a set of partial differential equation systems describing the Hall effect, which establishes the time evolution relation between the binormal electromagnetic field lines and magnetic vector potential in MHD. Finally, for a better comprehension of the time evolution of the binormal electromagnetic field lines, we consider the traveling wave transformation and the Riccati differential equation system to evaluate the evolution of the magnetic helicity and energy dissipation based on the calculation of the theoretical models. [ABSTRACT FROM AUTHOR]

This paper is intended to present a useful insight into the comprehension of the configurations of electromagneto-vortex formation and their implications in the magnetic helicity and energy dissipation in magnetohydrodynamics (MHD). We introduce the formulations of magnetic helicity of the Lorentz force of the Serret–Frenet frame vectors and associated electromagnetic vector fields E − B firstly. Then, we investigate the Godbillon–Vey helicity model and figure out the intrinsic geometric characterizations of the electromagnetic vector fields and magnetic vector potential under the influence of vanishing magnetic helicity. We also compute the time evolution of the electromagnetic field lines of the Frenet–Serret vectors and associated geometric quantities in MHD. Then, we determine a set of partial differential equation systems describing the Hall effect, which establishes the time evolution relation between the electromagnetic field lines and magnetic vector potential in MHD. Finally, for a better comprehension of the time evolution of the electromagnetic field lines, we improve an appropriate approach to evaluate the evolution of the magnetic helicity and energy dissipation based on calculations on theoretical models. [ABSTRACT FROM AUTHOR]

Motivated by recent researches in magnetic curves and their flows in different types of geometric manifolds and physical spacetime structures, we compute fractional Lorentz force equations associated with the magnetic n -lines in the normal direction in Minkowski space. Fractional evolution equations of magnetic n -lines due to inextensible Heisenberg antiferromagnetic flow are computed to construct the soliton surface associated with the inextensible Heisenberg antiferromagnetic flow. Then, their approximate solutions are investigated in terms of magnetic and geometric quantities via the conformable fractional derivative method. By considering arc-length and time-dependent orthogonal curvilinear coordinates, we finally determine the necessary and sufficient conditions that have to be satisfied by these quantities to define the Lorentz magnetic flux surfaces based on the inextensible Heisenberg antiferromagnetic flow model in Minkowski space. [ABSTRACT FROM AUTHOR]