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 $u(x,t) = \phi_\omega(x)e^{-i\omega t}$ 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.

The Vakhitov–Kolokolov stability criterion,{{ cite journal

|author=Колоколов, А. А.

|title=Устойчивость основной моды нелинейного волнового уравнения в кубичной среде

|journal=Прикладная механика и техническая физика

|issue=3

|year=1973

|pages=152–155

|url=https://www.sibran.ru/journals/issue.php?ID=156469&ARTICLE_ID=156604

}}

{{ cite journal

|author=A.A. Kolokolov

|title=Stability of the dominant mode of the nonlinear wave equation in a cubic medium

|journal=Journal of Applied Mechanics and Technical Physics

|volume=14

|issue=3

|year=1973

|pages=426–428

|doi=10.1007/BF00850963

|bibcode=1973JAMTP..14..426K

|s2cid=123792737

}}{{ cite journal

|author1=Вахитов, Н. Г. |author2=Колоколов, А. А.

|name-list-style=amp |title=Стационарные решения волнового уравнения в среде с насыщением нелинейности

|journal=Известия высших учебных заведений. Радиофизика

|volume=16

|year=1973

|pages=1020–1028 }}{{ cite journal

|author1=N.G. Vakhitov |author2=A.A. Kolokolov

|name-list-style=amp |title=Stationary solutions of the wave equation in the medium with nonlinearity saturation

|journal=Radiophys. Quantum Electron.

|volume=16

|issue=7

|year=1973

|pages=783–789

|doi=10.1007/BF01031343

|bibcode=1973R&QE...16..783V |s2cid=123386885

}}

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

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{{ cite journal

|author = Vladimir E. Zakharov

|title=Instability of Self-focusing of Light

|journal=Zh. Eksp. Teor. Fiz.

|year = 1967

|volume=53

|pages=1735–1743

|url=https://www.jetp.ac.ru/cgi-bin/dn/e_026_05_0994.pdf

|bibcode=1968JETP...26..994Z

}} by Vladimir Zakharov.

Generalizations

This result has been generalized to abstract Hamiltonian systems with U(1)-invariance.{{cite journal

|journal=J. Funct. Anal.

|volume=74

|year=1987

|pages=160–197

|doi=10.1016/0022-1236(87)90044-9|doi-access=free}}

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{{cite journal

|journal=Proceedings of the Royal Society A

|volume=411

|year=1987

|issue=1841

|pages=395–412

|doi=10.1098/rspa.1987.0073

|bibcode=1987RSPSA.411..395B |s2cid=120894859 }}

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.{{cite journal

|journal=J. Funct. Anal.

|volume=94

|issue=2 |year=1990

|pages=308–348

|doi=10.1016/0022-1236(90)90016-E |doi-access=free}}

References

Category:Stability theory

Category:Solitons

Vakhitov&ndash;Kolokolov stability criterion

Original formulation

Generalizations

See also

References

Vakhitov–Kolokolov stability criterion