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.

The LLE describes an anisotropic magnet. The equation is described in (Faddeev & Takhtajan 2007, chapter 8) as follows: It is an equation for a vector field S, in other words a function on R¹⁺ⁿ taking values in R³. The equation depends on a fixed symmetric 3 by 3 matrix J, usually assumed to be diagonal; that is, ${\displaystyle J=\operatorname {diag} (J_{1},J_{2},J_{3})}$. It is given by Hamilton's equation of motion for the Hamiltonian

(where J(S) is the quadratic form of J applied to the vector S) 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

In general case LLE (2) is nonintegrable. But it admits the two integrable reductions:

...
Wikipedia