Narrow Results

The subject of moving curves (and surfaces) in three-dimensional space (3-D) is a fascinating topic not only because it represents typical nonlinear dynamical systems in classical mechanics, but also finds important applications in a variety of physical problems in different disciplines. Making use of the underlying geometry, one can very often relate the associated evolution equations to many interesting nonlinear evolution equations, including soliton possessing nonlinear dynamical systems. Typical examples include dynamics of filament vortices in ordinary and superfluids, spin systems, phases in classical optics, various systems encountered in physics of soft matter, etc. Such interrelations between geometric evolution and physical systems have yielded considerable insight into the underlying dynamics. We present a succinct tutorial analysis of these developments in this article, and indicate further directions. We also point out how evolution equations for moving surfaces are often intimately related to soliton equations in higher dimensions.

In this paper, we relate the evolution equations of the electric field and magnetic field vectors of the polarized light ray traveling in a coiled optical fiber in the ordinary space into the nonlinear Schrödinger’s equation by proposing new kinds of binormal motions and new kinds of Hasimoto functions in addition to commonly known formula of the binormal motion and Hasimoto function. All these operations have been conducted by using the orthonormal frame of Bishop equations that is defined along with the coiled optical fiber. We also propose perturbed solutions of the nonlinear Schrödinger’s evolution equation that governs the propagation of solitons through the electric field $(E)$ and magnetic field $(M)$ vectors. Finally, we provide some numerical simulations to supplement the analytical outcomes.

In this paper, we give some constructions for the applications of optical magnetic Heisenberg spherical ferromagnetic chain of T - timelike magnetic particle by spherical de Sitter frame in de Sitter space. This aim may be concluded by well-known de Sitter frame or a new alternative spherical frame with an optical magnetic spherical Heisenberg ferromagnetic chain. Moreover, we achieve total magnetic phases of T - timelike magnetic particle evolutions. Finally, we obtain some numerical modeling of optical magnetic spherical Heisenberg ferromagnetic flows.