The Vakhitov–Kolokolov stability criterion is a condition for linear stability (sometimes called spectral stability) of solitary wave solutions to a wide class of 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 solitary wave
is the charge (or momentum) of the solitary wave
, conserved by Noether's theorem due to U(1)-invariance of the system.
Originally, this criterion was obtained for the nonlinear Schrödinger equation, where
is a smooth real-valued function.
Since the equation is U(1)-invariant, by Noether's theorem, it has an integral of motion,
, which is called charge or momentum, depending on the model under consideration.
For a wide class of functions
, the nonlinear Schrödinger equation admits solitary wave solutions of the form
decays for large
belongs to the Sobolev space
from an interval or collection of intervals of a real line.
The Vakhitov–Kolokolov stability criterion,[1][2][3][4] is a condition of spectral stability of a solitary wave solution.
Namely, if this condition is satisfied at a particular value of
, then the linearization at the solitary wave with this
This result is based on an earlier work[5] by Vladimir Zakharov.
This result has been generalized to abstract Hamiltonian systems with U(1)-invariance.
[6] It was shown that under rather general conditions the Vakhitov–Kolokolov stability criterion guarantees not only spectral stability but also orbital stability of solitary waves.
The stability condition has been generalized[7] to traveling wave solutions to the generalized Korteweg–de Vries equation of the form The stability condition has also been generalized to Hamiltonian systems with a more general symmetry group.