Landau–Lifshitz Model

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 R^1+''n'' taking values in R³. The equation depends on a fixed symmetric 3 by 3 matrix ''J'', usually assumed to be diagonal; that is, $J=\operatorname(J_, J_, J_)$. It is given by Hamilton's equation of motion for the Hamiltonian

:H=\frac\int \left sum_i\left(\frac\right)^-J(\mathbf)\right, dx\qquad (1)

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

:$\frac = \mathbf\wedge \sum_i\frac + \mathbf\wedge J\mathbf.\qquad (2)$

In 1+1 dimensions this equation is

:$\frac = \mathbf\wedge \frac + \mathbf\wedge J\mathbf.\qquad (3)$

In 2+1 dimensions this equation takes the form

:$\frac = \mathbf\wedge \left(\frac + \frac\right)+ \mathbf\wedge J\mathbf\qquad (4)$

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

:$\frac = \mathbf\wedge \left(\frac + \frac+\frac\right)+ \mathbf\wedge J\mathbf.\qquad (5)$

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 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.

{{DEFAULTSORT:Landau-Lifshitz model
Magnetic ordering
Partial differential equations
Lev Landau