Magnetic materials possess the intrinsic spin order, whose disturbance 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 dynamics 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 this 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 magnetic 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 constructed with approximation error depending quadratically on the strength of magnetic field and being uniform in time.

中文翻译：

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Landau–Lifshitz–Gilbert方程的自旋波解

磁性材料具有固有的自旋顺序，其干扰会导致自旋波。从数学角度看，自旋波被称为行波，通常在波动方程和输运方程中看到。内在自旋阶数的动力学是由Landau–Lifshitz–Gilbert方程建模的，Landau–Lifshitz–Gilbert方程是非线性抛物线方程组，具有点长度约束。在本文中，该方程式的自旋波是基于以下假设而获得的：自旋波在以不同速度传播时保持其在空间中的周期性。在没有磁场的情况下，提供了显式形式的自旋波。当施加磁场时，自旋波不具有这种明确的形式，但是其稳定性被严格证明是正确的。而且，