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Compacton
In the theory of integrable systems, a compacton, introduced in , is a soliton with compact support. An example of an equation with compacton solutions is the generalization : u_t+(u^m)_x+(u^n)_=0\, of the Korteweg–de Vries equation (KdV equation) with ''m'', ''n'' > 1. The case with ''m'' = ''n'' is the Rosenau–Hyman equation as used in their 1993 study; the case ''m'' = 2, ''n'' = 1 is essentially the KdV equation. Example The equation : u_t+(u^2)_x+(u^2)_=0 \, has a travelling wave solution given by : u(x,t) = \begin \dfrac\cos^2((x-\lambda t)/4) & \text, x - \lambda t, \le 2\pi, \\ \\ 0 & \text, x - \lambda t, \ge 2\pi. \end This has compact support in ''x'', and so is a compacton. See also * Peakon * Vector soliton References * * *{{citation, title=Exact discrete breather compactons in nonlinear Klein-Gordon lattices , last1=Comte , first1=Jean-Christophe , journal=Physical Review E ''Physical Review E'' is a pe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Soliton
In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium. (Dispersive effects are a property of certain systems where the speed of a wave depends on its frequency.) Solitons are the solutions of a widespread class of weakly nonlinear dispersive partial differential equations describing physical systems. The soliton phenomenon was first described in 1834 by John Scott Russell (1808–1882) who observed a solitary wave in the Union Canal in Scotland. He reproduced the phenomenon in a wave tank and named it the "Wave of Translation". Definition A single, consensus definition of a soliton is difficult to find. ascribe three properties to solitons: # They are of permanent form; # They are localized within a region; # They can interact with other solitons, and emerge from the collision unchanged, e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rosenau–Hyman Equation
The Rosenau–Hyman equation or ''K''(''n'',''n'') equation is a KdV-like equation having compacton solutions. This nonlinear 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 sol ... is of the form : u_t+a(u^n)_x+(u^n)_=0. \, The equation is named after Philip Rosenau and James M. Hyman, who used in their 1993 study of compactons. The ''K''(''n'',''n'') equation has the following traveling wave solutions: * when ''a'' > 0 :: u(x,t)= \left( \frac \sin^2 \left(\frac\sqrt(x-ct+b)\right)\right)^, *when ''a'' < 0 :: :: References [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Soliton
In physical optics or wave optics, a vector soliton is a solitary wave with multiple components coupled together that maintains its shape during propagation. Ordinary solitons maintain their shape but have effectively only one (scalar) polarization component, while vector solitons have two distinct polarization components. Among all the types of solitons, optical vector solitons draw the most attention due to their wide range of applications, particularly in generating ultrafast pulses and light control technology. Optical vector solitons can be classified into temporal vector solitons and spatial vector solitons. During the propagation of both temporal solitons and spatial solitons, despite being in a medium with birefringence, the orthogonal polarizations can copropagate as one unit without splitting due to the strong cross-phase modulation and coherent energy exchange between the two polarizations of the vector soliton which may induce intensity differences between these two pola ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integrable Systems
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, such that its behaviour has far fewer degrees of freedom than the dimensionality of its phase space; that is, its evolution is restricted to a submanifold within 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physical Review Letters
''Physical Review Letters'' (''PRL''), established in 1958, is a peer-reviewed, scientific journal that is published 52 times per year by the American Physical Society. As also confirmed by various measurement standards, which include the ''Journal Citation Reports'' impact factor and the journal ''h''-index proposed by Google Scholar, many physicists and other scientists consider ''Physical Review Letters'' to be one of the most prestigious journals in the field of physics. ''According to Google Scholar, PRL is the journal with the 9th journal h-index among all scientific journals'' ''PRL'' is published as a print journal, and is in electronic format, online and CD-ROM. Its focus is rapid dissemination of significant, or notable, results of fundamental research on all topics related to all fields of physics. This is accomplished by rapid publication of short reports, called "Letters". Papers are published and available electronically one article at a time. When published in s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Support
In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smallest closed set containing all points not mapped to zero. This concept is used very widely in mathematical analysis. Formulation Suppose that f : X \to \R is a real-valued function whose domain is an arbitrary set X. The of f, written \operatorname(f), is the set of points in X where f is non-zero: \operatorname(f) = \. The support of f is the smallest subset of X with the property that f is zero on the subset's complement. If f(x) = 0 for all but a finite number of points x \in X, then f is said to have . If the set X has an additional structure (for example, a topology), then the support of f is defined in an analogous way as the smallest subset of X of an appropriate type such that f vanishes in an appropriate sense on its complement. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Travelling Wave
In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (resting) value at some frequency. When the entire waveform moves in one direction, it is said to be a ''traveling wave''; by contrast, a pair of superimposed periodic waves traveling in opposite directions makes a ''standing wave''. In a standing wave, the amplitude of vibration has nulls at some positions where the wave amplitude appears smaller or even zero. Waves are often described by a ''wave equation'' (standing wave field of two opposite waves) or a one-way wave equation for single wave propagation in a defined direction. Two types of waves are most commonly studied in classical physics. In a ''mechanical wave'', stress and strain fields oscillate about a mechanical equilibrium. A mechanical wave is a local deformation (strain) in so ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peakon
In the theory of integrable systems, a peakon ("peaked soliton") is a soliton with discontinuous first derivative; the wave profile is shaped like the graph of the function e^. Some examples of non-linear partial differential equations with (multi-)peakon solutions are the Camassa–Holm shallow water wave equation, the Degasperis–Procesi equation and the Fornberg–Whitham equation. Since peakon solutions are only piecewise differentiable, they must be interpreted in a suitable weak sense. The concept was introduced in 1993 by Camassa and Holm in the short but much cited paper where they derived their shallow water equation. A family of equations with peakon solutions The primary example of a PDE which supports peakon solutions is : u_t - u_ + (b+1) u u_x = b u_x u_ + u u_, \, where u(x,t) is the unknown function, and ''b'' is a parameter. In terms of the auxiliary function m(x,t) defined by the relation m = u-u_, the equation takes the simpler form : m_t + m_x u + b m u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Notices Of The American Mathematical Society
''Notices of the American Mathematical Society'' is the membership journal of the American Mathematical Society (AMS), published monthly except for the combined June/July issue. The first volume appeared in 1953. Each issue of the magazine since January 1995 is available in its entirety on the journal web site. Articles are peer-reviewed by an editorial board of mathematical experts. Since 2019, the editor-in-chief is Erica Flapan. The cover regularly features mathematical visualization Mathematical phenomena can be understood and explored via visualization. Classically this consisted of two-dimensional drawings or building three-dimensional models (particularly plaster models in the 19th and early 20th century), while today it ...s. The ''Notices'' is self-described to be the world's most widely read mathematical journal. As the membership journal of the American Mathematical Society, the ''Notices'' is sent to the approximately 30,000 AMS members worldwide, one-third of whom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physical Review E
''Physical Review E'' is a peer-reviewed, scientific journal, published monthly by the American Physical Society. The main field of interest is collective phenomena of many-body systems. It is currently edited by Uwe C. Täuber. While original research content requires subscription, editorials, news, and other non-research content is openly accessible. Scope Although the focus of this journal is many-body phenomena, the broad scope of the journal includes quantum chaos, soft matter physics, classical chaos, biological physics and granular materials. Also emphasized are statistical physics, equilibrium and transport properties of fluids, liquid crystals, complex fluids, polymers, chaos, fluid dynamics, plasma physics, classical physics, and computational physics. About Physical Review E APS. July 2010 [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
