Landau–Lifshitz model
In solid-state physics, the Landau–Lifshitz equation (LLE), named for Lev Landau and Evgeny Lifshitz, is a partial differential equation describing time evolution of magnetism in solids, depending on 1 time variable and 1, 2, or 3 space variables. Landau–Lifshitz equation. The LLE describes an anisotropic magnet. The equation is described in as follows: It is an equation for a vector field S, in other words a function on R taking values in R. The equation depends on a fixed symmetric 3 by 3 matrix "J", usually assumed to be diagonal; that is, formula_1. It  is given by Hamilton's equation of motion for the Hamiltonian which is In 1+1 dimensions this equation is In 2+1 dimensions this equation takes the form which is the (2+1)-dimensional LLE. For the (3+1)-dimensional case LLE looks like Integrable reductions. In general case LLE (2) is nonintegrable. But it admits the two integrable reductions: