# Vakhitov–Kolokolov stability criterion

## From Wikipedia, the free encyclopedia

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 with frequency has the form

where is the 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 , , and is a smooth real-valued function. The solution is assumed to be complex-valued. 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
, where and decays for large
(one often requires that belongs to the Sobolev space ). Usually such solutions exist for 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 has no spectrum in the right half-plane.

This result is based on an earlier work^{[5]} by Vladimir Zakharov.

## Generalizations

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.^{[8]}

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