Landau–Lifshitz model#Landau–Lifshitz equation

The LLE describes an anisotropic magnet. The equation 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 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, $J=\backslash operatorname\{diag\}(J\_\{1\},\; J\_\{2\},\; J\_\{3\})$. The LLE is then given by Hamilton's equation of motion for the Hamiltonian

:$H=\backslash frac\{1\}\{2\}\backslash int\; \backslash left[\backslash sum\_i\backslash left(\backslash frac\{\backslash partial\; \backslash mathbf\{S\}\}\{\backslash partial\; x\_i\}\backslash right)^\{2\}-J(\backslash mathbf\{S\})\backslash right]\backslash ,\; dx\backslash qquad\; (1)$

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

which is

:$\backslash frac\{\backslash partial\; \backslash mathbf\{S\}\}\{\backslash partial\; t\}\; =\; \backslash mathbf\{S\}\backslash wedge\; \backslash sum\_i\backslash frac\{\backslash partial^2\; \backslash mathbf\{S\}\}\{\backslash partial\; x\_i^\{2\}\}\; +\; \backslash mathbf\{S\}\backslash wedge\; J\backslash mathbf\{S\}.\backslash qquad\; (2)$

In 1+1 dimensions, this equation is

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

In 2+1 dimensions, this equation takes the form

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

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

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

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.

• Nonlinear Schrödinger equation
• Heisenberg model (classical)
• Spin wave
• Micromagnetism
• Ishimori equation
• Magnet
• Ferromagnetism

• {{citation|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 | doi=10.1007/978-3-540-69969-9}}
• {{citation|ISBN= 978-981-277-875-8

|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.

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Category:Magnetic ordering

Category:Partial differential equations

Category:Lev Landau