Landau–Lifshitz model explained

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,

J = \operatorname{diag}(J _{1} , J _{2} , J _{3} ) . The LLE is then given by Hamilton's equation of motion for the HamiltonianH = (1)/(2) \int \left[\sum _{i\left( (\partial S)/(\partial x _{i})} \right) ^{2} - J( S )\right] dx ( 1 )

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

(\partial S)/(\partial t) = S \wedge \sum _{i (\partial ^{2} S)/(\partial x ^{2} _{i})} + S \wedge J S . ( 2 )

In 1+1 dimensions, this equation is

(\partial S)/(\partial t) = S \wedge (\partial ^{2} S)/(\partial x ^{2}) + S \wedge J S . ( 3 )

In 2+1 dimensions, this equation takes the form

(\partial S)/(\partial t) = S \wedge \left( (\partial ^{2} S)/(\partial x ^{2}) + (\partial ^{2} S)/(\partial y ^{2}) \right) + S \wedge J S ( 4 )

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

(\partial S)/(\partial t) = S \wedge \left( (\partial ^{2} S)/(\partial x ^{2}) + (\partial ^{2} S)/(+ \partial y ^{2}) (\partial ^{2} S)/(\partial z ^{2}) \right) + S \wedge J S . ( 5 )

Integrable reductions

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

a) in 1+1 dimensions, that is Eq. (3), it is integrable

b) 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.

See also

• Nonlinear Schrödinger equation
• Heisenberg model (classical)
• Spin wave
• Micromagnetism
• Ishimori equation
• Magnet
• Ferromagnetism

References

• • • Kosevich A.M., Ivanov B.A., Kovalev A.S. Nonlinear magnetization waves. Dynamical and topological solitons. – Kiev: Naukova Dumka, 1988. – 192 p.