Abstract
We consider the one-dimensional Landau–Lifshitz–Gilbert (LLG) equation, a model describing the dynamics for the spin in ferromagnetic materials. Our main aim is the analytical study of the bi-parametric family of self-similar solutions of this model. In the presence of damping, our construction provides a family of global solutions of the LLG equation which are associated with discontinuous initial data of infinite (total) energy, and which are smooth and have finite energy for all positive times. Special emphasis will be given to the behaviour of this family of solutions with respect to the Gilbert damping parameter.We would like to emphasize that our analysis also includes the study of self-similar solutions of the Schrödinger map and the heat flow for harmonic maps into the 2-sphere as special cases. In particular, the results presented here recover some of the previously known results in the setting of the 1D-Schrödinger map equation.
Highlights
Introduction and statement of resultsIn this work we consider the one-dimensional Landau–Lifshitz–Gilbert equation (LLG)∂tm = βm × mss − αm × (m × mss), s ∈ R, t > 0, (LLG)where m = (m1, m2, m3) : R × (0, ∞) −→ S2 is the spin vector, β ≥ 0, α ≥ 0, × denotes the usual cross-product in R3, and S2 is the unit sphere in R3.Here we have not included the effects of anisotropy or an external magnetic field
One of the main goals of this paper is to study both the qualitative and quantitative effect of the damping parameter α and the parameter c0 on the dynamical behaviour of the family {mc0,α}c0,α of self-similar solutions of (LLG) found in Theorem 1.1
The approach we use in the proof of the main results in this paper is based on the integration of the Serret–Frenet system of equations via a Riccati equation, which in turn can be reduced to the study of a second order ordinary differential equation given by f ′′(s)
Summary
In this work we consider the one-dimensional Landau–Lifshitz–Gilbert equation (LLG). where m = (m1, m2, m3) : R × (0, ∞) −→ S2 is the spin vector, β ≥ 0, α ≥ 0, × denotes the usual cross-product in R3, and S2 is the unit sphere in R3. Has a jump singularity at the point s = 0 whenever the vectors A+c0,α and A−c0,α satisfy In this situation (and we will be able to prove analytically this is the case at least for certain ranges of the parameters α and c0, see Proposition 1.5 below), Theorem 1.1 provides a bi-parametric family of global smooth solutions of (LLG) associated to a discontinuous singular initial data (jump-singularity). It is important to remark that in the setting of Schrödinger equations, for fixed α ∈ [0, 1] and c0 > 0, the solution mc0,α(s, t) of (LLG) established in Theorem 1.1 is associated through the Hasimoto transformation (1.2) to the filament function uc0,α(s, t).
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
