The Cauchy problem of generalized Landau-Lifshitz equation into S n

In this paper, we study the Cauchy problem of an integrable evolution system, i.e., the n-dimensional generalization of third-order symmetry of the well-known Landau-Lifshitz equation. By rewriting this equation in a geometric form and applying the geometric energy method with a forth-order perturbation, we show the global well-posedness of the Cauchy problem in suitable Sobolev spaces.