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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.

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 **R**^{1+n} taking values in **R**^{3}. 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

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.

(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

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.

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

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.

**Continuum mechanics** is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such models in the 19th century.

In physics, the **Navier–Stokes equations**, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of viscous fluid substances.

In mathematics, the **Hodge star operator** or **Hodge star** is a linear map introduced by W. V. D. Hodge. It is defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. The result when applied to an element of the algebra is called the element's **Hodge dual**.

The **finite volume method** (**FVM**) is a method for representing and evaluating partial differential equations in the form of algebraic equations [LeVeque, 2002; Toro, 1999]. Similar to the finite difference method or finite element method, values are calculated at discrete places on a meshed geometry. "Finite volume" refers to the small volume surrounding each node point on a mesh. In the finite volume method, volume integrals in a partial differential equation that contain a divergence term are converted to surface integrals, using the divergence theorem. These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods are conservative. Another advantage of the finite volume method is that it is easily formulated to allow for unstructured meshes. The method is used in many computational fluid dynamics packages.

The **Classical Heisenberg model** is the case of the n-vector model, one of the models used in statistical physics to model ferromagnetism, and other phenomena.

In continuum mechanics, the **finite strain theory**—also called **large strain theory**, or **large deformation theory**—deals with deformations in which strains and/or rotations are large enough to invalidate assumptions inherent in infinitesimal strain theory. In this case, the undeformed and deformed configurations of the continuum are significantly different, requiring a clear distinction between them. This is commonly the case with elastomers, plastically-deforming materials and other fluids and biological soft tissue.

**Aeroacoustics** is a branch of acoustics that studies noise generation via either turbulent fluid motion or aerodynamic forces interacting with surfaces. Noise generation can also be associated with periodically varying flows. A notable example of this phenomenon is the Aeolian tones produced by wind blowing over fixed objects.

In physics, **Hamilton's principle** is William Rowan Hamilton's formulation of the principle of stationary action. It states that the dynamics of a physical system is determined by a variational problem for a functional based on a single function, the Lagrangian, which contains all physical information concerning the system and the forces acting on it. The variational problem is equivalent to and allows for the derivation of the *differential* equations of motion of the physical system. Although formulated originally for classical mechanics, Hamilton's principle also applies to classical fields such as the electromagnetic and gravitational fields, and plays an important role in quantum mechanics, quantum field theory and criticality theories.

The **Navier–Stokes existence and smoothness** problem concerns the mathematical properties of solutions to the Navier–Stokes equations, one of the pillars of fluid mechanics. These equations describe the motion of a fluid in space. Solutions to the Navier–Stokes equations are used in many practical applications. However, theoretical understanding of the solutions to these equations is incomplete. In particular, solutions of the Navier–Stokes equations often include turbulence, which remains one of the greatest unsolved problems in physics, despite its immense importance in science and engineering.

The **London equations**, developed by brothers Fritz and Heinz London in 1935, relate current to electromagnetic fields in and around a superconductor. Arguably the simplest meaningful description of superconducting phenomena, they form the genesis of almost any modern introductory text on the subject. A major triumph of the equations is their ability to explain the Meissner effect, wherein a material exponentially expels all internal magnetic fields as it crosses the superconducting threshold.

There are various **mathematical descriptions of the electromagnetic field** that are used in the study of electromagnetism, one of the four fundamental interactions of nature. In this article, several approaches are discussed, although the equations are in terms of electric and magnetic fields, potentials, and charges with currents, generally speaking.

In isotropic turbulence the **Kármán–Howarth** equation, which is derived from the Navier–Stokes equations, is used to describe the evolution of non-dimensional longitudinal autocorrelation.

The intent of this article is to highlight the important points of the **derivation of the Navier–Stokes equations** as well as its application and formulation for different families of fluids.

In fluid dynamics, the **Taylor–Green vortex** is an unsteady flow of a decaying vortex, which has an exact closed form solution of the incompressible Navier–Stokes equations in Cartesian coordinates. It is named after the British physicist and mathematician Geoffrey Ingram Taylor and his collaborator A. E. Green.

In physics, the **Landau–Lifshitz–Gilbert equation**, named for Lev Landau, Evgeny Lifshitz, and T. L. Gilbert, is a name used for a differential equation describing the precessional motion of magnetization **M** in a solid. It is a modification by Gilbert of the original equation of Landau and Lifshitz.

**Lagrangian mechanics** is a reformulation of classical mechanics, introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in 1788.

The **fractional Schrödinger equation** is a fundamental equation of fractional quantum mechanics. It was discovered by Nick Laskin (1999) as a result of extending the Feynman path integral, from the Brownian-like to Lévy-like quantum mechanical paths. The term *fractional Schrödinger equation* was coined by Nick Laskin.

- 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.

The **International Standard Book Number** (**ISBN**) is a numeric commercial book identifier which is intended to be unique. Publishers purchase ISBNs from an affiliate of the International ISBN Agency.

* Mathematical Reviews* is a journal published by the American Mathematical Society (AMS) that contains brief synopses, and in some cases evaluations, of many articles in mathematics, statistics, and theoretical computer science. The AMS also publishes an associated online bibliographic database called MathSciNet which contains an electronic version of

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