Nonlinear Partial Differential Equations in Physics and Mechanics

Abstract

Let Ω be a domain in m-dimensional Enclidean space R ^m. The n-dimensional classical system for the isotropic Heisenberg chain with spin density Z _i (x, t) (i = 1, 2, 3) is described by the Hamiltonian density

$$ H = \frac{{{{\alpha }_{2}}}}{2}{{\left| {\nabla \vec{Z}} \right|}^{2}} - {{H}_{0}}\vec{Z} $$

where α₂ is the exchange constant, $\vec{Z}$ is the spin veetor and H ₀ an external magnetie field. The spin equation of motion with Gilbert damping term (without the external magnetic field) has the form

$$ {{\partial }_{t}}\vec{Z} = - {{\alpha }_{1}}\vec{Z} \times (\vec{Z} \times \Delta \vec{Z}) + {{\alpha }_{2}}\vec{Z} \times \Delta \vec{Z}, $$

((0.1))

where “ ×” denotes the vector cross product in ${{R}^{3}},\vec{Z} = ({{Z}_{1}},{{Z}_{2}},{{Z}_{3}}):\Omega \times [0,T] \to {{R}^{3}}$ is the spin vector and α₁≥0 is a Gilbert damping constant (see [1]). The system (0.1) is implied by the conservation of energy and magnitude of $\vec{Z}$, and is aversion which gives rise to a continuum spin wave theory.

1991 Mathematics Subject Classification: 35Q

Supported by the National Natural Science Fund of China