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AKNS System
In mathematics, the AKNS system is an integrable system of partial differential equations, introduced by and named after Mark J. Ablowitz, David J. Kaup, Alan C. Newell, and Harvey Segur from their publication in Studies in Applied Mathematics: . Definition The AKNS system is a pair of two partial differential equations for two complex-valued functions ''p'' and ''q'' of 2 variables ''t'' and ''x'': : p_t=+ip^2q-\fracp_ : q_t=-iq^2p+\fracq_ If ''p'' and ''q'' are complex conjugates this reduces to the nonlinear Schrödinger equation. Huygens' principle applied to the Dirac operator gives rise to the AKNS hierarchy. Applications to General Relativity In October of 2021, the dynamics of three-dimensional (extremal) black holes on General Relativity with negative cosmological constant were shown equivalent to two independent copies of the AKNS system. This duality was addressed through the imposition of suitable boundary conditions to the Chern-Simons action. In this sche ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integrable System
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first integrals, that its motion is confined to a submanifold of much smaller dimensionality than that of its phase space. Three features are often referred to as characterizing integrable systems: * the existence of a ''maximal'' set of conserved quantities (the usual defining property of complete integrability) * the existence of algebraic invariants, having a basis in algebraic geometry (a property known sometimes as algebraic integrability) * the explicit determination of solutions in an explicit functional form (not an intrinsic property, but something often referred to as solvability) Integrable systems may be seen as very different in qualitative character from more ''generic'' dynamical systems, which are more typically chaotic syste ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differential Equations
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to how is thought of as an unknown number solving, e.g., an algebraic equation like . However, it is usually impossible to write down explicit formulae for solutions of partial differential equations. There is correspondingly a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity and stability. Among the many open questions are the existence an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mark J
Mark may refer to: In the Bible * Mark the Evangelist (5–68), traditionally ascribed author of the Gospel of Mark * Gospel of Mark, one of the four canonical gospels and one of the three synoptic gospels Currencies * Mark (currency), a currency or unit of account in many nations * Bosnia and Herzegovina convertible mark, the currency of Bosnia and Herzegovina * East German mark, the currency of the German Democratic Republic * Estonian mark, the currency of Estonia between 1918 and 1928 * Finnish markka (), the currency of Finland from 1860 until 28 February 2002 * Polish mark (), the currency of the Kingdom of Poland and of the Republic of Poland between 1917 and 1924 German * Deutsche Mark, the official currency of West Germany from 1948 until 1990 and later the unified Germany from 1990 until 2002 * German gold mark, the currency used in the German Empire from 1873 to 1914 * German Papiermark, the German currency from 4 August 1914 * German rentenmark, a currency iss ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alan C
Alan may refer to: People *Alan (surname), an English and Kurdish surname * Alan (given name), an English given name ** List of people with given name Alan ''Following are people commonly referred to solely by "Alan" or by a homonymous name.'' * Alan (Chinese singer) (born 1987), female Chinese singer of Tibetan ethnicity, active in both China and Japan * Alan (Mexican singer) (born 1973), Mexican singer and actor *Alan (wrestler) (born 1975), a.k.a. Gato Eveready, who wrestles in Asistencia Asesoría y Administración * Alan (footballer, born 1979) (Alan Osório da Costa Silva), Brazilian footballer * Alan (footballer, born 1998) (Alan Cardoso de Andrade), Brazilian footballer *Alan I, King of Brittany (died 907), "the Great" * Alan II, Duke of Brittany (c. 900–952) * Alan III, Duke of Brittany(997–1040) * Alan IV, Duke of Brittany (c. 1063–1119), a.k.a. Alan Fergant ("the Younger" in Breton language) * Alan of Tewkesbury, 12th century abbott * Alan of Lynn (c. 1348–1423) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Studies In Applied Mathematics
The journal ''Studies in Applied Mathematics'' is published by Wiley–Blackwell on behalf of the Massachusetts Institute of Technology. It features scholarly articles on mathematical applications in allied fields, notably computer science, mechanics, astrophysics, geophysics, biophysics, and high-energy physics. Its pedigree came from the ''Journal of Mathematics and Physics'' which was founded by the MIT Mathematics Department in 1920. The Journal changed to its present name in 1969. The journal was edited from 1969 by David Benney of the Department of Mathematics, Massachusetts Institute of Technology. According to ISI Journal Citation Reports ''Journal Citation Reports'' (''JCR'') is an annual publication by Clarivate. It has been integrated with the Web of Science and is accessed from the Web of Science Core Collection. It provides information about academic journals in the natur ..., in 2020, it ranked 26th among the 265 journals in the Applied Mathematics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex-valued Function
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, and applied mathematics, as well as in physics, including the branches of hydrodynamics, thermodynamics, quantum mechanics, and twistor theory. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering. As a differentiable function of a complex variable is equal to the sum function given by its Taylor series (that is, it is analytic), complex analysis is particularly concerned with analytic functions of a complex variable, that is, ''holomorphic functions''. The concept can be extended to functions of several complex variables. Complex analysis is contrasted with real analysis, which deals with th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a and b are real numbers, then the complex conjugate of a + bi is a - bi. The complex conjugate of z is often denoted as \overline or z^*. In polar form, if r and \varphi are real numbers then the conjugate of r e^ is r e^. This can be shown using Euler's formula. The product of a complex number and its conjugate is a real number: a^2 + b^2 (or r^2 in polar coordinates). If a root of a univariate polynomial with real coefficients is complex, then its complex conjugate is also a root. Notation The complex conjugate of a complex number z is written as \overline z or z^*. The first notation, a vinculum, avoids confusion with the notation for the conjugate transpose of a matrix, which can be thought of as a generalization of the complex conjugate. The second is preferred in physics, where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonlinear Schrödinger Equation
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, planar waveguides and hot rubidium vapors 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 me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirac Operator
In mathematics and in quantum mechanics, a Dirac operator is a first-order differential operator that is a formal square root, or half-iterate, of a second-order differential operator such as a Laplacian. It was introduced in 1847 by William Hamilton and in 1928 by Paul Dirac. The question which concerned Dirac was to factorise formally the Laplace operator of the Minkowski space, to get an equation for the wave function which would be compatible with special relativity. Formal definition In general, let ''D'' be a first-order differential operator acting on a vector bundle ''V'' over a Riemannian manifold ''M''. If : D^2=\Delta, \, where ∆ is the (positive, or geometric) Laplacian of ''V'', then ''D'' is called a Dirac operator. Note that there are two different conventions as to how the Laplace operator is defined: the "analytic" Laplacian, which could be characterized in \R^n as \Delta=\nabla^2=\sum_^n\Big(\frac\Big)^2 (which is negative-definite, in the sense that \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
