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 u ( x , t ) = ϕ ω ( x ) e − i ω t {\displaystyle u(x,t)=\phi _{\omega }(x)e^{-i\omega t}} with frequency ω {\displaystyle \omega } has the form d d ω Q ( ω ) < 0 , {\displaystyle {\frac {d}{d\omega }}Q(\omega )<0,} where Q ( ω ) {\displaystyle Q(\omega )\,} is the charge (or momentum) of the solitary wave ϕ ω ( x ) e − i ω t {\displaystyle \phi _{\omega }(x)e^{-i\omega t}} , conserved by Noether's theorem due to U(1)-invariance of the system.

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