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Vakhitov–Kolokolov Stability Criterion
The Vakhitov–Kolokolov stability criterion is a stability criterion, condition for linear stability (sometimes called ''spectral stability'') of soliton, solitary wave solutions to a wide class of unitary invariance, U(1)-invariant Hamiltonian systems, named after Soviet scientists Aleksandr Kolokolov (Александр Александрович Колоколов) and Nazib Vakhitov (Назиб Галиевич Вахитов). The condition for linear stability of a soliton, solitary wave u(x,t) = \phi_\omega(x)e^ with frequency \omega has the form : \fracQ(\omega)<0, where is the electric charge, charge (or momentum) of the solitary wave , conserved by Noether's theorem due to U(1)-invariance of the system. Original formulation Originally, this criterion was obtained for the nonlinear Schrödinger equation, : where |
Stability Criterion
In control theory, and especially stability theory, a stability criterion establishes when a system is stable polynomial, stable. A number of stability criteria are in common use: *Circle criterion *Jury stability criterion *Liénard–Chipart criterion *Nyquist stability criterion *Routh–Hurwitz stability criterion *Vakhitov–Kolokolov stability criterion *Barkhausen stability criterion Stability may also be determined by means of root locus analysis. Although the concept of stability is general, there are several narrower definitions through which it may be assessed: * BIBO stability * Linear stability * Lyapunov stability * Orbital stability {{sia Stability theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sobolev Space
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, i.e. a Banach space. Intuitively, a Sobolev space is a space of functions possessing sufficiently many derivatives for some application domain, such as partial differential equations, and equipped with a norm that measures both the size and regularity of a function. Sobolev spaces are named after the Russian mathematician Sergei Sobolev. Their importance comes from the fact that weak solutions of some important partial differential equations exist in appropriate Sobolev spaces, even when there are no strong solutions in spaces of continuous functions with the derivatives understood in the classical sense. Motivation In this section and throughout the article \Omega is an open subset of \R^n. There are man ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orbital Stability
In mathematical physics and the theory of partial differential equations, the solitary wave solution of the form u(x,t)=e^\phi(x) is said to be orbitally stable if any solution with the initial data sufficiently close to \phi(x) forever remains in a given small neighborhood of the trajectory of e^\phi(x). Formal definition Formal definition is as follows. Consider the dynamical system : i\frac=A(u), \qquad u(t)\in X, \quad t\in\R, with X a Banach space over \Complex, and A : X \to X. We assume that the system is \mathrm(1)-invariant, so that A(e^u) = e^A(u) for any u\in X and any s\in\R. Assume that \omega \phi=A(\phi), so that u(t)=e^\phi is a solution to the dynamical system. We call such solution a solitary wave. We say that the solitary wave e^\phi is orbitally stable if for any \epsilon > 0 there is \delta > 0 such that for any v_0\in X with \Vert \phi-v_0\Vert_X < \delta there is a solution defined for all such that |
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] |
Lyapunov Stability
Various types of stability may be discussed for the solutions of differential equations or difference equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. This may be discussed by the theory of Aleksandr Lyapunov. In simple terms, if the solutions that start out near an equilibrium point x_e stay near x_e forever, then x_e is Lyapunov stable. More strongly, if x_e is Lyapunov stable and all solutions that start out near x_e converge to x_e, then x_e is said to be ''asymptotically stable'' (see asymptotic analysis). The notion of '' exponential stability'' guarantees a minimal rate of decay, i.e., an estimate of how quickly the solutions converge. The idea of Lyapunov stability can be extended to infinite-dimensional manifolds, where it is known as structural stability, which concerns the behavior of different but "nearby" solutions to differential equations. Input-to-state stability (ISS ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Stability
In mathematics, in the theory of differential equations and dynamical systems, a particular stationary or quasistationary solution to a nonlinear system is called linearly unstable if the linearization of the equation at this solution has the form dr/dt = A r, where ''r'' is the perturbation to the steady state, ''A'' is a linear operator whose spectrum contains eigenvalues with ''positive'' real part. If all the eigenvalues have ''negative'' real part, then the solution is called linearly stable. Other names for linear stability include exponential stability or stability in terms of first approximation. If there exists an eigenvalue with ''zero'' real part then the question about stability cannot be solved on the basis of the first approximation and we approach the so-called "centre and focus problem". Examples Ordinary differential equation The differential equation \frac = x - x^2 has two stationary (time-independent) solutions: ''x'' = 0 and ''x'' = 1. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derrick's Theorem
Derrick's theorem is an argument by physicist G. H. Derrick which shows that stationary localized solutions to a nonlinear wave equation or nonlinear Klein–Gordon equation in spatial dimensions three and higher are unstable. Original argument Derrick's paper, which was considered an obstacle to interpreting soliton-like solutions as particles, contained the following physical argument about non-existence of stable localized stationary solutions to the nonlinear wave equation :\nabla^2 \theta-\frac=\frac 1 2 f'(\theta), \qquad \theta(x,t)\in\R,\quad x\in\R^3, now known under the name of Derrick's Theorem. (Above, f(s) is a differentiable function with f'(0)=0.) The energy of the time-independent solution \theta(x)\, is given by : E=\int\left \nabla\theta)^2+f(\theta)\right\, d^3 x. A necessary condition for the solution to be stable is \delta^2 E\ge 0\,. Suppose \theta(x)\, is a localized solution of \delta E=0\,. Define \theta_\lambda(x)=\theta(\lambda x)\, where \lamb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetry Group
In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient space which takes the object to itself, and which preserves all the relevant structure of the object. A frequent notation for the symmetry group of an object ''X'' is ''G'' = Sym(''X''). For an object in a metric space, its symmetries form a subgroup of the isometry group of the ambient space. This article mainly considers symmetry groups in Euclidean geometry, but the concept may also be studied for more general types of geometric structure. Introduction We consider the "objects" possessing symmetry to be geometric figures, images, and patterns, such as a wallpaper pattern. For symmetry of physical objects, one may also take their physical composition as part of the pattern. (A pattern may be specified formally as a scalar field, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Korteweg–de Vries Equation
A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. Generalizations posit the existence of a domain or set of elements, as well as one or more common characteristics shared by those elements (thus creating a conceptual model). As such, they are the essential basis of all valid deductive inferences (particularly in logic, mathematics and science), where the process of verification is necessary to determine whether a generalization holds true for any given situation. Generalization can also be used to refer to the process of identifying the parts of a whole, as belonging to the whole. The parts, which might be unrelated when left on their own, may be brought together as a group, hence belonging to the whole by establishing a common relation between them. However, the parts cannot be generalized into a whole—until a common relation is established among ''all'' parts. This does not mean that the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proceedings Of The Royal Society A
''Proceedings of the Royal Society'' is the main research journal of the Royal Society. The journal began in 1831 and was split into two series in 1905: * Series A: for papers in physical sciences and mathematics. * Series B: for papers in life sciences. Many landmark scientific discoveries are published in the Proceedings, making it one of the most important science journals in history. The journal contains several articles written by prominent scientists such as Paul Dirac, Werner Heisenberg, Ernest Rutherford, Erwin Schrödinger, William Lawrence Bragg, Lord Kelvin, J.J. Thomson, James Clerk Maxwell, Dorothy Hodgkin and Stephen Hawking. In 2004, the Royal Society began '' The Journal of the Royal Society Interface'' for papers at the interface of physical sciences and life sciences. History The journal began in 1831 as a compilation of abstracts of papers in the '' Philosophical Transactions of the Royal Society'', the older Royal Society publication, that began in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orbital Stability
In mathematical physics and the theory of partial differential equations, the solitary wave solution of the form u(x,t)=e^\phi(x) is said to be orbitally stable if any solution with the initial data sufficiently close to \phi(x) forever remains in a given small neighborhood of the trajectory of e^\phi(x). Formal definition Formal definition is as follows. Consider the dynamical system : i\frac=A(u), \qquad u(t)\in X, \quad t\in\R, with X a Banach space over \Complex, and A : X \to X. We assume that the system is \mathrm(1)-invariant, so that A(e^u) = e^A(u) for any u\in X and any s\in\R. Assume that \omega \phi=A(\phi), so that u(t)=e^\phi is a solution to the dynamical system. We call such solution a solitary wave. We say that the solitary wave e^\phi is orbitally stable if for any \epsilon > 0 there is \delta > 0 such that for any v_0\in X with \Vert \phi-v_0\Vert_X < \delta there is a solution defined for all such that |
Vladimir E
Vladimir (, , pre-1918 orthography: ) is a masculine given name of Slavic origin, widespread throughout all Slavic nations in different forms and spellings. The earliest record of a person with the name is Vladimir of Bulgaria (). Etymology The Old East Slavic form of the name is Володимѣръ ''Volodiměr'', while the Old Church Slavonic form is ''Vladiměr''. According to Max Vasmer, the name is composed of Slavic владь ''vladĭ'' "to rule" and ''*mēri'' "great", "famous" (related to Gothic element ''mērs'', ''-mir'', cf. Theode''mir'', Vala''mir''). The modern ( pre-1918) Russian forms Владимиръ and Владиміръ are based on the Church Slavonic one, with the replacement of мѣръ by миръ or міръ resulting from a folk etymological association with миръ "peace" or міръ "world". Max Vasmer, ''Etymological Dictionary of Russian Language'' s.v. "Владимир"starling.rinet.ru [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
