Vakhitov–Kolokolov stability criterion


The Vakhitov–Kolokolov stability criterion is a condition for linear stability of solitary wave solutions to a wide class of U-invariant Hamiltonian systems, named after Soviet scientists Aleksandr Kolokolov and Nazib Vakhitov.
The condition for linear stability of a solitary wave with frequency has the form
where is the charge of the solitary wave
conserved by Noether's theorem due to U-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-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
.
Usually such solutions exist for from an interval or collection of intervals
of a real line.
Vakhitov–Kolokolov stability criterion,
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 by Vladimir Zakharov.

Generalizations

This result has been generalized to abstract Hamiltonian systems with U-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
The stability condition has also been generalized
to Hamiltonian systems with a more general symmetry group