Landau–Lifshitz model

For another Landau–Lifshitz equation describing magnetism, see Landau–Lifshitz–Gilbert equation.

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 2007, chapter 8) 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, $J=\operatorname {diag} (J_{1},J_{2},J_{3})$. The LLE is then given by Hamilton's equation of motion for the Hamiltonian

$H={\frac {1}{2}}\int \left[\sum _{i}\left({\frac {\partial \mathbf {S} }{\partial x_{i}}}\right)^{2}-J(\mathbf {S} )\right]\,dx\qquad (1)$

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

${\frac {\partial \mathbf {S} }{\partial t}}=\mathbf {S} \wedge \sum _{i}{\frac {\partial ^{2}\mathbf {S} }{\partial x_{i}^{2}}}+\mathbf {S} \wedge J\mathbf {S} .\qquad (2)$

In 1+1 dimensions, this equation is

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

In 2+1 dimensions, this equation takes the form

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

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

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

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

• 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, MR 2348643
• 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.