Landau–Lifshitz model

Landau–Lifshitz equation

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, J = diag ⁡ ( J 1 , J 2 , J 3 ) . It is given by Hamilton's equation of motion for the Hamiltonian

H = 1 2 ∫ [ ∑ i ( ∂ S ∂ x i ) 2 − J ( S ) ] d x ( 1 )

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

∂ S ∂ t = S ∧ ∑ i ∂ 2 S ∂ x i 2 + S ∧ J S . ( 2 )

In 1+1 dimensions this equation is

∂ S ∂ t = S ∧ ∂ 2 S ∂ x 2 + S ∧ J S . ( 3 )

In 2+1 dimensions this equation takes the form

∂ S ∂ t = S ∧ ( ∂ 2 S ∂ x 2 + ∂ 2 S ∂ y 2 ) + S ∧ J S ( 4 )

which is the (2+1)-dimensional LLE. For the (3+1)-dimensional case LLE looks like

∂ S ∂ t = S ∧ ( ∂ 2 S ∂ x 2 + ∂ 2 S ∂ y 2 + ∂ 2 S ∂ z 2 ) + S ∧ J S . ( 5 )

Integrable reductions

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

a) in the 1+1 dimensions, that is Eq. (3), it is integrableb) when J = 0 . In this case the (1+1)-dimensional LLE (3) turns into the continuous classical Heisenberg ferromagnet equation (see e.g. Heisenberg model (classical)) which is already integrable.