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 formula_1 with frequency formula_2 has the form
formula_3
where formula_4 is the charge (or momentum) of the solitary wave
formula_5,
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,
formula_6
where formula_7, formula_8, and formula_9 is a smooth real-valued function. The solution formula_10 is assumed to be complex-valued. Since the equation is U(1)-invariant, by Noether's theorem, it has an integral of motion,
formula_11, which is called charge or momentum, depending on the model under consideration.
For a wide class of functions formula_12, the nonlinear Schrödinger equation admits solitary wave solutions of the form
formula_1, where formula_14 and formula_15 decays for large formula_16
(one often requires that formula_15 belongs to the Sobolev space formula_18). Usually such solutions exist for formula_2 from an interval or collection of intervals of a real line.
The Vakhitov–Kolokolov stability criterion,
formula_20
is a condition of spectral stability of a solitary wave solution. Namely, if this condition is satisfied at a particular value of formula_2, then the linearization at the solitary wave with this formula_2 has no spectrum in the right half-plane.
This result is based on an earlier work by Vladimir Zakharov.
Generalizations.
This result has been generalized to abstract Hamiltonian systems with U(1)-invariance.
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
to traveling wave solutions
to the generalized Korteweg–de Vries equation of the form
formula_23.
The stability condition has also been generalized to Hamiltonian systems with a more general symmetry group.