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 system
A Hamiltonian system is a dynamical system governed by Hamilton's equations. In physics, this dynamical system describes the evolution of a physical system such as a planetary system or an electron in an electromagnetic field. These systems can ...
s, 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
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass ...
) of the solitary wave
,
conserved by
Noether's theorem
Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether ...
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
Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether ...
, it has an
integral of motion In mechanics, a constant of motion is a quantity that is conserved throughout the motion, imposing in effect a constraint on the motion. However, it is a ''mathematical'' constraint, the natural consequence of the equations of motion, rather than ...
,
, which is called
charge or
momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass ...
, 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
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense ...
). Usually such solutions exist for
from an
interval or collection of intervals of a real line.
The Vakhitov–Kolokolov stability criterion,
:
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 by
Vladimir Zakharov.
Generalizations
This result has been generalized to abstract
Hamiltonian system
A Hamiltonian system is a dynamical system governed by Hamilton's equations. In physics, this dynamical system describes the evolution of a physical system such as a planetary system or an electron in an electromagnetic field. These systems can ...
s with U(1)-invariance.
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
to traveling wave solutions
to the
generalized Korteweg–de Vries equation
A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. Generalizations posit the existence of a domain or set of elements, as well as one or more common characteri ...
of the form
:
.
The stability condition has also been generalized to Hamiltonian systems with a more general
symmetry group
In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...
.
See also
*
Derrick's theorem Derrick's theorem is an argument by physicist G. H. Derrick
which shows that stationary localized solutions to a nonlinear wave equation
or nonlinear Klein–Gordon equation
in spatial dimensions three and higher are unstable.
Original argum ...
*
Linear stability
*
Lyapunov stability
*
Nonlinear Schrödinger equation
*
Orbital stability
References
{{DEFAULTSORT:Vakhitov-Kolokolov stability criterion
Stability theory
Solitons