Vakhitov–Kolokolov stability criterion explained

u(x,t) = \phi ^{- i\omega t} _{\omega(x)e} with frequency \omega has the form(d)/(d\omega) Q(\omega)< 0 , where Q(\omega) is the charge (or momentum) of the solitary wave\phi ^{- i\omega t} _{\omega(x)e} ,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 (\partial)/(\partial t) u(x,t) =- (\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,$Q(u)\; =\; \backslash frac\; \backslash int\_|u(x,t)|^2\backslash ,dx$, 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 formu(x,t) = \phi ^{- i\omega t} _{\omega(x)e} , 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.The Vakhitov–Kolokolov stability criterion,^{[1] [2] [3] [4]} (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^[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 stabilitycriterion guarantees not only spectral stabilitybut also orbital stability of solitary waves.

The stability condition has been generalized^[7] to traveling wave solutionsto the generalized Korteweg–de Vries equation of the form

\partial _{t} u + \partial ^{3} _{x} 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

References

1. Колоколов, А. А.. Устойчивость основной моды нелинейного волнового уравнения в кубичной среде. Прикладная механика и техническая физика. 3. 1973. 152–155.
2. A.A. Kolokolov. Stability of the dominant mode of the nonlinear wave equation in a cubic medium. Journal of Applied Mechanics and Technical Physics. 14. 3. 1973. 426–428. 10.1007/BF00850963. 1973JAMTP..14..426K. 123792737.
3. Вахитов, Н. Г. . Колоколов, А. А. . amp . Стационарные решения волнового уравнения в среде с насыщением нелинейности. Известия высших учебных заведений. Радиофизика. 16. 1973. 1020–1028 .
4. N.G. Vakhitov . A.A. Kolokolov. amp . Stationary solutions of the wave equation in the medium with nonlinearity saturation. Radiophys. Quantum Electron.. 16. 7. 1973. 783–789. 10.1007/BF01031343. 1973R&QE...16..783V . 123386885.
5. Vladimir E. Zakharov. Instability of Self-focusing of Light. Zh. Eksp. Teor. Fiz.. 1967. 53. 1735–1743. 1968JETP...26..994Z.
6. Manoussos Grillakis . Jalal Shatah . Walter Strauss . amp . Stability theory of solitary waves in the presence of symmetry. I. J. Funct. Anal.. 74. 1987. 160–197. 10.1016/0022-1236(87)90044-9. free.
7. Jerry Bona . Panagiotis Souganidis . Walter Strauss . amp . Stability and instability of solitary waves of Korteweg-de Vries type. Proceedings of the Royal Society A. 411. 1987. 1841. 395–412. 10.1098/rspa.1987.0073. 1987RSPSA.411..395B . 120894859 .
8. Manoussos Grillakis . Jalal Shatah . Walter Strauss . amp . Stability theory of solitary waves in the presence of symmetry. J. Funct. Anal.. 94. 2 . 1990. 308–348. 10.1016/0022-1236(90)90016-E . free.