# Orbital stability

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In mathematical physics and the 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 .

## Formal definition

Formal definition is as follows.^{[1]} 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

According to ^{[2]}
,^{[3]}
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,^{[4]}^{[5]}
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 sufficiently large.

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