Vakhitov–Kolokolov stability criterion

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 ${\displaystyle u(x,t)=\phi _{\omega }(x)e^{-i\omega t}}$ with frequency ${\displaystyle \omega }$ has the form

${\displaystyle {\frac {d}{d\omega }}Q(\omega )<0,}$

where ${\displaystyle Q(\omega )\,}$ is the charge (or momentum) of the solitary wave ${\displaystyle \phi _{\omega }(x)e^{-i\omega t}}$, conserved by Noether's theorem due to U(1)-invariance of the system.