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 Russian scientists Aleksandr Kolokolov (Александр Александрович Колоколов) and Nazib Vakhitov (Назиб Галиевич Вахитов). The condition for linear stability of a solitary wave $u(x,t)=\phi_\omega(x)e^{-i\omega t}\,$ with frequency $\omega\,$ has the form

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

where $Q(\omega)\,$ is the charge (or momentum) of the solitary wave $\phi_\omega(x)e^{-i\omega t}\,$, 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,

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

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

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

is a condition of spectral stability of a solitary wave solution. Namely, if this condition is satisfied at a particular value of $\omega\,$, then the linearization at the solitary wave with this $\omega\,$ has no spectrum in the right half-plane.

This result is based on an earlier work^[3] by Vladimir Zakharov.

Generalizations

This result has been generalized to abstract Hamiltonian systems with U(1)-invariance .^[4] 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 ^[5] to traveling wave solutions to the generalized Korteweg–de Vries equation of the form

$\partial_t u + \partial_x^3 u + \partial_x f(u) = 0\,$.

The stability condition has also been generalized to Hamiltonian systems with a more general symmetry group .^[6]

See also

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

References

1. Вахитов, Н. Г. and Колоколов, А. А. (1973). "Стационарные решения волнового уравнения в среде с насыщением нелинейности". Известия высших учебных заведений. Радиофизика 16: 1020–1028.
2. N.G. Vakhitov and A.A. Kolokolov (1973). "Stationary solutions of the wave equation in the medium with nonlinearity saturation". Radiophys. Quantum Electron. 16: 783–789. Bibcode:1973R&QE...16..783V. doi:10.1007/BF01031343.
3. Vladimir E. Zakharov (1967). "Instability of Self-focusing of Light" (PDF). Zh. Eksp. Teor. Fiz 53: 1735–1743. Bibcode:1968JETP...26..994Z.
4. Manoussos Grillakis, Jalal Shatah, and Walter Strauss (1987). "Stability theory of solitary waves in the presence of symmetry. I". J. Funct. Anal. 74: 160–197. doi:10.1016/0022-1236(87)90044-9.
5. Jerry Bona, Panagiotis Souganidis, and Walter Strauss (1987). "Stability and instability of solitary waves of Korteweg-de Vries type". Proceedings of the Royal Society A 411 (1841): 395–412. Bibcode:1987RSPSA.411..395B. doi:10.1098/rspa.1987.0073.
6. Manoussos Grillakis, Jalal Shatah, and Walter Strauss (1990). "Stability theory of solitary waves in the presence of symmetry". J. Funct. Anal. 94: 308–348. doi:10.1016/0022-1236(90)90016-E.