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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] |
