# 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.

Solid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state physics studies how the large-scale properties of solid materials result from their atomic-scale properties. Thus, solid-state physics forms a theoretical basis of materials science. It also has direct applications, for example in the technology of transistors and semiconductors.

Lev Davidovich Landau was a Soviet physicist who made fundamental contributions to many areas of theoretical physics.

Evgeny Mikhailovich Lifshitz was a leading Soviet physicist and brother of the physicist Ilya Lifshitz.

## Landau–Lifshitz equation

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, ${\displaystyle J=\operatorname {diag} (J_{1},J_{2},J_{3})}$. It is given by Hamilton's equation of motion for the Hamiltonian

In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. A vector field in the plane, can be visualised as: a collection of arrows with a given magnitude and direction, each attached to a point in the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from one point to another point.

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. For example, the dimensions of the matrix below are 2 × 3, because there are two rows and three columns:

In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero. The term usually refers to square matrices. An example of a 2-by-2 diagonal matrix is ; the following matrix is a 3-by-3 diagonal matrix: . An identity matrix of any size, or any multiple of it, will be a diagonal matrix.

${\displaystyle H={\frac {1}{2}}\int \left[\sum _{i}\left({\frac {\partial \mathbf {S} }{\partial x_{i}}}\right)^{2}-J(\mathbf {S} )\right]\,dx\qquad (1)}$

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

${\displaystyle {\frac {\partial \mathbf {S} }{\partial t}}=\mathbf {S} \wedge \sum _{i}{\frac {\partial ^{2}\mathbf {S} }{\partial x_{i}^{2}}}+\mathbf {S} \wedge J\mathbf {S} .\qquad (2)}$

In 1+1 dimensions this equation is

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

In 2+1 dimensions this equation takes the form

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

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

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

## 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
b) when ${\displaystyle J=0}$. 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.

In theoretical physics, the (one-dimensional) nonlinear Schrödinger equation (NLSE) is a nonlinear variation of the Schrödinger equation. It is a classical field equation whose principal applications are to the propagation of light in nonlinear optical fibers and planar waveguides and to Bose-Einstein condensates confined to highly anisotropic cigar-shaped traps, in the mean-field regime. Additionally, the equation appears in the studies of small-amplitude gravity waves on the surface of deep inviscid (zero-viscosity) water; the Langmuir waves in hot plasmas; the propagation of plane-diffracted wave beams in the focusing regions of the ionosphere; the propagation of Davydov's alpha-helix solitons, which are responsible for energy transport along molecular chains; and many others. More generally, the NLSE appears as one of universal equations that describe the evolution of slowly varying packets of quasi-monochromatic waves in weakly nonlinear media that have dispersion. Unlike the linear Schrödinger equation, the NLSE never describes the time evolution of a quantum state. The 1D NLSE is an example of an integrable model.

Spin waves are propagating disturbances in the ordering of magnetic materials. These low-lying collective excitations occur in magnetic lattices with continuous symmetry. From the equivalent quasiparticle point of view, spin waves are known as magnons, which are bosonic modes of the spin lattice that correspond roughly to the phonon excitations of the nuclear lattice. As temperature is increased, the thermal excitation of spin waves reduces a ferromagnet's spontaneous magnetization. The energies of spin waves are typically only μeV in keeping with typical Curie points at room temperature and below. The discussion of spin waves in antiferromagnets is beyond the scope of this article.

The Ishimori equation (IE) is a partial differential equation proposed by the Japanese mathematician Ishimori (1984). Its interest is as the first example of a nonlinear spin-one field model in the plane that is integrable Sattinger, Tracy & Venakides.

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## 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, 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.

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