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, . 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