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 (Faddeev Takhtajan) 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, $J=\mathrm{diag}(J_1,J_2,J_3)$. It is given by Hamilton's equation of motion for the Hamiltonian

$H=\frac{1}{2}\int [\sum _i{\left(\frac{\partial \mathbf{𝐒}}{\partial x_i}\right)}^2-J(\mathbf{𝐒})]dx(1)$

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

$\frac{\partial \mathbf{𝐒}}{\partial t}=\mathbf{𝐒}\wedge \sum _i\frac{{\partial }^2\mathbf{𝐒}}{\partial x_i^2}+\mathbf{𝐒}\wedge J\mathbf{𝐒}.(2)$

In 1+1 dimensions this equation is

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

In 2+1 dimensions this equation takes the form

$\frac{\partial \mathbf{𝐒}}{\partial t}=\mathbf{𝐒}\wedge (\frac{{\partial }^2\mathbf{𝐒}}{\partial x^2}+\frac{{\partial }^2\mathbf{𝐒}}{\partial y^2})+\mathbf{𝐒}\wedge J\mathbf{𝐒}(4)$

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

$\frac{\partial \mathbf{𝐒}}{\partial t}=\mathbf{𝐒}\wedge (\frac{{\partial }^2\mathbf{𝐒}}{\partial x^2}+\frac{{\partial }^2\mathbf{𝐒}}{\partial y^2}+\frac{{\partial }^2\mathbf{𝐒}}{\partial z^2})+\mathbf{𝐒}\wedge J\mathbf{𝐒}.(5)$

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 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

• Faddeev, Ludwig D.; Takhtajan, Leon A. (2007), Hamiltonian methods in the theory of solitons, Classics in Mathematics, Berlin: Springer, pp. x+592, doi:10.1007/978-3-540-69969-9, ISBN 978-3-540-69843-2 
• Guo, Boling; Ding, Shijin (2008), Landau-Lifshitz Equations, Frontiers of Research With the Chinese Academy of Sciences, World Scientific Publishing Company, ISBN 978-981-277-875-8 
• Kosevich A.M., Ivanov B.A., Kovalev A.S. Nonlinear magnetization waves. Dynamical and topological solitons. – Kiev: Naukova Dumka, 1988. – 192 p.