Lamb has identified a certain class of moving space curves with soliton equations. We show that there are two other classes of curve evolution that may be so identified. Hence three distinct classes of curve evolution are associated with a given integrable equation. The nonlinear Schrödinger equation is used to illustrate this explicitly.

Using the complex stereographic variable representation for the macrospin, from a study of the nonlinear dynamics underlying the generalized Landau-Lifshitz(LL) equation with Gilbert damping, we show that the spin-transfer torque is effectively equivalent to an applied magnetic field. We study the macrospin switching on a Stoner particle due to spin-transfer torque on application of a spin polarized current. We find that the switching due to spin-transfer torque is a more effective alternative to switching by an applied external field in the presence of damping. We demonstrate numerically that a spin-polarized current in the form of a short pulse can be effectively employed to achieve the desired macro-spin switching.

We study the bifurcation and chaos scenario of the macromagnetization vector in a homogeneous nanoscale-ferromagnetic thin film of the type used in spin-valve pillars. The underlying dynamics is described by a generalized Landau-Lifshitz-Gilbert (LLG) equation. The LLG equation has an especially appealing form under a complex stereographic projection, wherein the qualitative equivalence of an applied field and a spin-current induced torque is transparent. Recently, chaotic behavior of such a spin vector has been identified by Li et al. [ Phys. Rev. B 74, 054417 (2006)] using a spin-polarized current passing through the pillar of constant polarization direction and periodically varying magnitude, owing to the spin-transfer torque effect. In this paper, we show that the same dynamical behavior can be achieved using a periodically varying applied magnetic field in the presence of a constant dc magnetic field and constant spin current, which is technically much more feasible, and demonstrate numerically the chaotic dynamics in the system for an infinitely thin film. Further, it is noted that in the presence of a nonzero crystal anisotropy field, chaotic dynamics occurs at much lower magnitudes of the spin current and dc applied field.

We apply our recent formalism establishing new connections between the geometry of moving space curves and soliton equations, to the nonlinear Schrödinger equation (NLS). We show that any given solution of the NLS gets associated with three distinct space curve evolutions. The tangent vector of the first of these curves, the binormal vector of the second and the normal vector of the third, are shown to satisfy the integrable Landau-Lifshitz (LL) equation S u = S × S ss , (S 2 = 1). These connections enable us to find the three surfaces swept out by the moving curves associated with the NLS. As an example, surfaces corresponding to a stationary envelope soliton solution of the NLS are obtained.

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 1 equations for moving surfaces are often intimately related to soliton equations in higher dimensions.