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 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,. The LLE is then 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, the LLE looks like

Integrable reductions

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