The Landau-Lifshitz equation of the ferromagnetic spin chain and harmonic maps
Let \(M\) be a Riemannian manifold without boundary and \(N\) be \(S^ 2\). The aim of the paper is to solve the Landau-Lifshitz (LL) type equations of \(M\) into \(S^ 2\): \[ \partial_ t u= -\alpha_ 1 u(u\Delta_ M u)+ \alpha_ 2 u\Delta_ M u, \] \((\alpha_ 1>0\), \(\alpha_ 2=\text{const})\). A global existence of solutions for the LL-equation is proved and some new links between harmonic maps and the solutions of the LL-equation of the ferromagnetic spin chain are established.
scientific article; zbMATH DE number 509960