A SPIN-WAVE SOLUTION TO THE LANDAU-LIFSHITZ-GILBERT EQUATION

Magnetic materials possess the intrinsic spin order, whose distrubance leads to spin waves. From the mathematical perspective, a spin wave is known as a traveling wave, which is often seen in wave and transport equations. The dynmics of intrinsic spin order is modeled by the Landau-Lifshitz-Gilbert equation, a nonlinear parabolic system of equations with a pointwise length constraint. In this paper, a spin wave for thid equation is obtained based on the assumption that the spin wave maintains its periodicity in space when propagating at a varying velocity. In the absence of magnetic field, an explicit form of spin wave is provided. When a magnetc field is applied, the spin wave does not have such an explicit form but its stability is justified rigorously. Moreover, an approximate explicit solution is construsted with approximation error depending quadraticaalt on the strength of magnetic field being uniform in time.