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 (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 (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, $J=\operatorname {diag} (J_{1},J_{2},J_{3})$. It is 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 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 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) turns into the continuous classical Heisenberg ferromagnet equation (see e.g. Heisenberg model (classical)) which is, as is well known, 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, ISBN 978-3-540-69843-2, MR2348643
• Guo, Boling; Ding, Shijin (2008), Landau-Lifshitz Equations, Frontiers of Research With the Chinese Academy of Sciences, World Scientific Publishing Company, ISBN 978-9812778758
• Kosevich A.M., Ivanov B.A., Kovalev A.S. Nonlinear magnetization waves. Dynamical and topological solitons.- Kiev: Naukova Dumka, 1988. - 192 p.