Landau–Lifshitz model

In solid-state physics, the Landau-Lifshitz equation (LLE), named for Lev Landau and Evgeny Lifshitz, is a partial differential equations describing time evolution of magnetism in solids, depending on 1 time variable and 1, 2, or 3 space variables.

Landau-Lifshitz equation

The LLE (a partial differential equation in 1 time and "n" space variables ("n" is usually 1,2,3) describes an anisotropic magnet.The LLE is described in harv|Faddeev|Takhtajan|2007|loc=chapter 8 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, $J=operatorname\{diag\}(J\_\{1\},\; J\_\{2\},\; J\_\{3\})$. It is given by Hamilton's equation of motion for the Hamiltonian

:$H=frac\{1\}\{2\}int\; left\; [sum\_ileft(frac\{partial\; mathbf\{S\{partial\; x\_i\}\; ight)^\{2\}-J(mathbf\{S\})\; ight]\; ,\; dxqquad\; (1)$

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

:$frac\{partial\; mathbf\{S\{partial\; t\}\; =\; mathbf\{S\}wedge\; sum\_ifrac\{partial^2\; mathbf\{S\{partial\; x\_i^\{2\; +\; mathbf\{S\}wedge\; Jmathbf\{S\}.qquad\; (2)$

In 1+1 dimensions this equation is

:$frac\{partial\; mathbf\{S\{partial\; t\}\; =\; mathbf\{S\}wedge\; frac\{partial^2\; mathbf\{S\{partial\; x^\{2\; +\; mathbf\{S\}wedge\; Jmathbf\{S\}.qquad\; (3)$

In 2+1 dimensions this equation takes the form

:$frac\{partial\; mathbf\{S\{partial\; t\}\; =\; mathbf\{S\}wedge\; left(frac\{partial^2\; mathbf\{S\{partial\; x^\{2\; +\; frac\{partial^2\; mathbf\{S\{partial\; y^\{2\; ight)+\; mathbf\{S\}wedge\; Jmathbf\{S\}qquad\; (4)$

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

:$frac\{partial\; mathbf\{S\{partial\; t\}\; =\; mathbf\{S\}wedge\; left(frac\{partial^2\; mathbf\{S\{partial\; x^\{2\; +\; frac\{partial^2\; mathbf\{S\{partial\; y^\{2+frac\{partial^2\; mathbf\{S\{partial\; z^\{2\; ight)+\; mathbf\{S\}wedge\; Jmathbf\{S\}.qquad\; (5)$

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. The Lax representation for this case reads as:$???qquad\; (6a)$:$???qquad\; (6b)$: b) if $J=0$ then the (1+1)-dimensional LLE (3) turn to the continuous classical Heisenberg ferromagnet equation (see e.g. Heisenberg model (classical)) which is as well known integrable.

ee also

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

References

*citation|id=MR|2348643|last= Faddeev|first= Ludwig D.|last2= Takhtajan|first2= Leon A.|title= Hamiltonian methods in the theory of solitons|series= Classics in Mathematics|publisher= Springer|place= Berlin|year= 2007|pages= x+592 | ISBN= 978-3-540-69843-2
*citation|ISBN= 978-9812778758
title=Landau-Lifshitz Equations |series=Frontiers of Research With the Chinese Academy of Sciences
first=Boling |last=Guo |first2= Shijin |last2=Ding|year=2008
publisher=World Scientific Publishing Company
* Kosevich A.M., Ivanov B.A., Kovalev A.S. Nonlinear magnetization waves. Dynamical and topological solitons.- Kiev: Naukova Dumka, 1988. - 192 p.