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 $Q(\backslash omega)\backslash ,$ is the electric charge, charge (or momentum) of the solitary wave $\backslash phi\_\backslash omega(x)e^$, 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, :$i\backslash fracu(x,t)=\; -\backslash frac\; u(x,t)\; +g(,\; u(x,t),\; ^2)u(x,t),$ where
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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 ...
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