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Vakhitov–Kolokolov stability criterion

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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 with frequency has the form

where is the charge (or momentum) of the solitary wave , 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,

where , , and is a smooth real-valued function. The solution is assumed to be complex-valued. Since the equation is U(1)-invariant, by Noether's theorem, it has an integral of motion, , which is called charge or momentum, depending on the model under consideration. For a wide class of functions , the nonlinear Schrödinger equation admits solitary wave solutions of the form , where and decays for large (one often requires that belongs to the Sobolev space ). Usually such solutions exist for from an interval or collection of intervals of a real line. The Vakhitov–Kolokolov stability criterion,[1][2][3][4]

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

This result is based on an earlier work[5] by Vladimir Zakharov.


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


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

See also


  1. ^ Колоколов, А. А. (1973). "Устойчивость основной моды нелинейного волнового уравнения в кубичной среде". Прикладная механика и техническая физика (3): 152–155.
  2. ^ A.A. Kolokolov (1973). "Stability of the dominant mode of the nonlinear wave equation in a cubic medium". Journal of Applied Mechanics and Technical Physics. 14 (3): 426–428. Bibcode:1973JAMTP..14..426K. doi:10.1007/BF00850963.
  3. ^ Вахитов, Н. Г. & Колоколов, А. А. (1973). "Стационарные решения волнового уравнения в среде с насыщением нелинейности". Известия высших учебных заведений. Радиофизика. 16: 1020–1028.
  4. ^ N.G. Vakhitov & A.A. Kolokolov (1973). "Stationary solutions of the wave equation in the medium with nonlinearity saturation". Radiophys. Quantum Electron. 16 (7): 783–789. Bibcode:1973R&QE...16..783V. doi:10.1007/BF01031343.
  5. ^ Vladimir E. Zakharov (1967). "Instability of Self-focusing of Light" (PDF). Zh. Eksp. Teor. Fiz. 53: 1735–1743. Bibcode:1968JETP...26..994Z.
  6. ^ Manoussos Grillakis; Jalal Shatah & 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.
  7. ^ Jerry Bona; Panagiotis Souganidis & 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.
  8. ^ Manoussos Grillakis; Jalal Shatah & Walter Strauss (1990). "Stability theory of solitary waves in the presence of symmetry". J. Funct. Anal. 94 (2): 308–348. doi:10.1016/0022-1236(90)90016-E.
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Vakhitov–Kolokolov stability criterion
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