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 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 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 Nonlinear Schrödinger equation Heisenberg model (classical) Spin wave Micromagnetism Ishimori equation Magnet Ferromagnetism
References
Kosevich A.M., Ivanov B.A., Kovalev A.S. Nonlinear magnetization waves. Dynamical and topological solitons. – Kiev: Naukova Dumka, 1988. – 192 p.