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

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". 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.