 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 realvalued 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. 160197.
 ^ 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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