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.

Original formulation

Originally, this criterion was obtained for the nonlinear Schrödinger equation,

${\displaystyle i{\frac {\partial }{\partial t}}u(x,t)=-{\frac {\partial ^{2}}{\partial x^{2}}}u(x,t)+g(|u(x,t)|^{2})u(x,t),}$

where ${\displaystyle x\in \mathbb {R} }$, ${\displaystyle t\in \mathbb {R} }$, and ${\displaystyle g\in C^{\infty }(\mathbb {R} )}$ is a smooth real-valued function. The solution ${\displaystyle u(x,t)}$ is assumed to be complex-valued. Since the equation is U(1)-invariant, by Noether's theorem, it has an integral of motion, ${\textstyle Q(u)={\frac {1}{2}}\int _{\mathbb {R} }|u(x,t)|^{2}\,dx}$, which is called charge or momentum, depending on the model under consideration. For a wide class of functions ${\displaystyle g}$, the nonlinear Schrödinger equation admits solitary wave solutions of the form ${\displaystyle u(x,t)=\phi _{\omega }(x)e^{-i\omega t}}$, where ${\displaystyle \omega \in \mathbb {R} }$ and ${\displaystyle \phi _{\omega }(x)}$ decays for large ${\displaystyle x}$ (one often requires that ${\displaystyle \phi _{\omega }(x)}$ belongs to the Sobolev space ${\displaystyle H^{1}(\mathbb {R} ^{n})}$). Usually such solutions exist for ${\displaystyle \omega }$ from an interval or collection of intervals of a real line. The Vakhitov–Kolokolov stability criterion,^[1][2][3][4]

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

is a condition of spectral stability of a solitary wave solution. Namely, if this condition is satisfied at a particular value of ${\displaystyle \omega }$, then the linearization at the solitary wave with this ${\displaystyle \omega }$ 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

${\displaystyle \partial _{t}u+\partial _{x}^{3}u+\partial _{x}f(u)=0\,}$.

The stability condition has also been generalized to Hamiltonian systems with a more general symmetry group.^[8]

See also

• Derrick's theorem
• Linear stability
• Lyapunov stability
• Nonlinear Schrödinger equation
• Orbital stability