- Orbital stability
-
In mathematical physics or theory of partial differential equations, the solitary wave solution of the form
is said to be orbitally stable if any solution with the initial data sufficiently close to
forever remains in a given small neighborhood of the trajectory of
.
Contents
Formal definition
Formal definition is as follows[1]. Let us consider the dynamical system
with
a Banach space over
, and
. We assume that the system is -invariant, so that
for any
and any
.
Assume that
, so that
is a solution to the dynamical system. We call such solution a solitary wave.
We say that the solitary wave
is orbitally stable if for any
there is
such that for any
with
there is a solution
defined for all
such that
, and such that this solution satisfies
Example
The solitary wave solution
to the nonlinear Schrödinger equation
where
is a smooth real-valued function, is orbitally stable if the Vakhitov–Kolokolov stability criterion is satisfied:
where
is the charge of the solution
, which is conserved in time (at least if the solution
is sufficiently smooth).
It was also shown [2] that if
at a particular value of
, then the solitary wave
is Lyapunov stable, with the Lyapunov function given by
, where
is the energy of a solution
, with
the antiderivative of
, as long as the constant
is chosen sufficienty large.
See also
References
- ^ Manoussos Grillakis, Jalal Shatah, Walter Strauss, Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal. 74 (1987), pp. 160--197.
- ^ Richard Jordan and Bruce Turkington, Statistical equilibrium theories for the nonlinear Schrödinger equation, Advances in wave interaction and turbulence (South Hadley, MA, 2000), Contemp. Math. 283 (2001), pp. 27–39.
Categories:- Stability theory
- Solitons
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