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.