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Dispersive Partial Differential Equation
In mathematics, a dispersive partial differential equation or dispersive PDE is a partial differential equation that is dispersive. In this context, dispersion means that waves of different wavelength propagate at different phase velocities. Examples Linear equations * Euler–Bernoulli beam equation with time-dependent loading *Airy equation *Schrödinger equation *Klein–Gordon equation Nonlinear equations *nonlinear Schrödinger equation * Korteweg–de Vries equation (or KdV equation) *Boussinesq equation (water waves) * sine–Gordon equation See also *Dispersion (optics) *Dispersion (water waves) *Dispersionless equation Dispersionless (or quasi-classical) limits of integrable partial differential equations (PDE) arise in various problems of mathematics and physics and have been intensively studied in recent literature (see e.g. references below). They typically a ... External links *ThDispersive PDE Wiki {{mathanalysis-stub Partial differential equations Nonlin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...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 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. Unl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dispersionless Equation
Dispersionless (or quasi-classical) limits of integrable partial differential equations (PDE) arise in various problems of mathematics and physics and have been intensively studied in recent literature (see e.g. references below). They typically arise when considering slowly modulated long waves of an integrable dispersive PDE system. Examples Dispersionless KP equation The dispersionless Kadomtsev–Petviashvili equation (dKPE), also known (up to an inessential linear change of variables) as the Khokhlov–Zabolotskaya equation, has the form : (u_t+uu_)_x+u_=0,\qquad (1) It arises from the commutation : _1, L_20.\qquad (2) of the following pair of 1-parameter families of vector fields : L_1=\partial_y+\lambda\partial_x-u_x\partial_,\qquad (3a) : L_2=\partial_t+(\lambda^2+u)\partial_x+(-\lambda u_x+u_y)\partial_,\qquad (3b) where \lambda is a spectral parameter. The dKPE is the x-dispersionless limit of the celebrated Kadomtsev–Petviashvili equation, arising when considering ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dispersion (water Waves)
In fluid dynamics, dispersion of water waves generally refers to frequency dispersion, which means that waves of different wavelengths travel at different phase speeds. Water waves, in this context, are waves propagating on the water surface, with gravity and surface tension as the restoring forces. As a result, water with a free surface is generally considered to be a dispersive medium. For a certain water depth, surface gravity waves – i.e. waves occurring at the air–water interface and gravity as the only force restoring it to flatness – propagate faster with increasing wavelength. On the other hand, for a given (fixed) wavelength, gravity waves in deeper water have a larger phase speed than in shallower water. In contrast with the behavior of gravity waves, capillary waves (i.e. only forced by surface tension) propagate faster for shorter wavelengths. Besides frequency dispersion, water waves also exhibit amplitude dispersion. This is a nonlinear effect, by which wa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dispersion (optics)
In optics, and by analogy other branches of physics dealing with wave propagation, dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency; sometimes the term chromatic dispersion is used for specificity to optics in particular. A medium having this common property may be termed a dispersive medium (plural ''dispersive media''). Although the term is used in the field of optics to describe light and other electromagnetic waves, dispersion in the same sense can apply to any sort of wave motion such as acoustic dispersion in the case of sound and seismic waves, and in gravity waves (ocean waves). Within optics, dispersion is a property of telecommunication signals along transmission lines (such as microwaves in coaxial cable) or the pulses of light in optical fiber. Physically, dispersion translates in a loss of kinetic energy through absorption. In optics, one important and familiar consequence of dispersion is the change in the angle of refra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sine–Gordon Equation
The sine-Gordon equation is a nonlinear hyperbolic partial differential equation in 1 + 1 dimensions involving the d'Alembert operator and the sine of the unknown function. It was originally introduced by in the course of study of surfaces of constant negative curvature as the Gauss–Codazzi equation for surfaces of curvature −1 in 3-space, and rediscovered by in their study of crystal dislocations known as the Frenkel–Kontorova model. This equation attracted a lot of attention in the 1970s due to the presence of soliton solutions. Origin of the equation and its name There are two equivalent forms of the sine-Gordon equation. In the (real) ''space-time coordinates'', denoted (''x'', ''t''), the equation reads: : \varphi_ - \varphi_ + \sin\varphi = 0, where partial derivatives are denoted by subscripts. Passing to the light-cone coordinates (''u'', ''v''), akin to ''asymptotic coordinates'' where : u = \frac, \quad v = \frac, the equation takes the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boussinesq Equation (water Waves)
In fluid dynamics, the Boussinesq approximation for water waves is an approximation valid for weakly non-linear and fairly long waves. The approximation is named after Joseph Boussinesq, who first derived them in response to the observation by John Scott Russell of the wave of translation (also known as solitary wave or soliton). The 1872 paper of Boussinesq introduces the equations now known as the Boussinesq equations. The Boussinesq approximation for water waves takes into account the vertical structure of the horizontal and vertical flow velocity. This results in non-linear partial differential equations, called Boussinesq-type equations, which incorporate frequency dispersion (as opposite to the shallow water equations, which are not frequency-dispersive). In coastal engineering, Boussinesq-type equations are frequently used in computer models for the simulation of water waves in shallow seas and harbours. While the Boussinesq approximation is applicable to fairly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Klein–Gordon Equation
The Klein–Gordon equation (Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a relativistic wave equation, related to the Schrödinger equation. It is second-order in space and time and manifestly Lorentz-covariant. It is a quantized version of the relativistic energy–momentum relation E^2 = (pc)^2 + \left(m_0c^2\right)^2\,. Its solutions include a quantum scalar or pseudoscalar field, a field whose quanta are spinless particles. Its theoretical relevance is similar to that of the Dirac equation. Electromagnetic interactions can be incorporated, forming the topic of scalar electrodynamics, but because common spinless particles like the pions are unstable and also experience the strong interaction (with unknown interaction term in the Hamiltonian,) the practical utility is limited. The equation can be put into the form of a Schrödinger equation. In this form it is expressed as two coupled differential equations, each of first order in time. The so ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differential Equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to how is thought of as an unknown number to be solved for in an algebraic equation like . However, it is usually impossible to write down explicit formulas for solutions of partial differential equations. There is, correspondingly, a vast amount of modern mathematical and scientific research on methods to Numerical methods for partial differential equations, numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematics, 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 a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schrödinger Equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the subject. The equation is named after Erwin Schrödinger, who postulated the equation in 1925, and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933. Conceptually, the Schrödinger equation is the quantum counterpart of Newton's second law in classical mechanics. Given a set of known initial conditions, Newton's second law makes a mathematical prediction as to what path a given physical system will take over time. The Schrödinger equation gives the evolution over time of a wave function, the quantum-mechanical characterization of an isolated physical system. The equation can be derived from the fact that the time-evolution operator must be unitary, and must therefore be generated by t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Airy Equation
In the physical sciences, the Airy function (or Airy function of the first kind) is a special function named after the British astronomer George Biddell Airy (1801–1892). The function and the related function , are linearly independent solutions to the differential equation \frac - xy = 0 , known as the Airy equation or the Stokes equation. This is the simplest second-order linear differential equation with a turning point (a point where the character of the solutions changes from oscillatory to exponential). Definitions For real values of ''x'', the Airy function of the first kind can be defined by the improper Riemann integral: \operatorname(x) = \dfrac\int_0^\infty\cos\left(\dfrac + xt\right)\, dt\equiv \dfrac \lim_ \int_0^b \cos\left(\dfrac + xt\right)\, dt, which converges by Dirichlet's test. For any real number x there is positive real number M such that function \dfrac3 + xt is increasing, unbounded and convex with continuous and unbounded derivative on interval lin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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