Landau–Lifshitz model

Template:For 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 Template:Harv 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, $J=\mathrm{diag}(J_1,J_2,J_3)$. The LLE is then given by Hamilton's equation of motion for the Hamiltonian

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

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

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

In 1+1 dimensions, this equation is

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

In 2+1 dimensions, this equation takes the form

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

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

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

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

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