Landau–Lifshitz model


In solid-state physics, the Landau–Lifshitz equation, 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,. 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 -dimensional LLE. For the -dimensional case LLE looks like

Integrable reductions

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