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 Russian scientists Aleksandr Kolokolov (Александр Александрович Колоколов) and Nazib Vakhitov (Назиб Галиевич Вахитов). The condition for linear stability of a solitary wave
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where
Original formulation
Originally, this criterion was obtained for the nonlinear Schrödinger equation,
where
is a condition of spectral stability of a solitary wave solution. Namely, if this condition is satisfied at a particular value of
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
The stability condition has also been generalized to Hamiltonian systems with a more general symmetry group .