Landau–Lifshitz model

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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

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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,  . The LLE is then 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, the LLE looks like

 

Integrable reductions

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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  . 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

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References

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  • 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.