“Landau–Lifshitz model”的意思、由来-开放百科全书

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Landau–Lifshitz model

释义

Landau–Lifshitz equation
Integrable reductions
See also
References

{{For|another Landau–Lifshitz equation describing magnetism|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 {{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, . It is given by Hamilton's equation of motion for the Hamiltonian

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

which is

In 1+1 dimensions this equation is

In 2+1 dimensions this equation takes the form

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

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 . 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.

References

{{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.

2 : Magnetic ordering|Partial differential equations

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Landau–Lifshitz equation

Integrable reductions

See also

References