Orbital stability

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 is as follows.

[1] Consider the dynamical system with

a Banach space over

We assume that the system is

ω ϕ =

( ϕ )

− i ω t

ϕ

is a solution to the dynamical system.

We call such solution a solitary wave.

We say that the solitary wave

− i ω t

ϕ

is orbitally stable if for any

‖ ϕ −

there is a solution

, and such that this solution satisfies According to [2] ,[3] the solitary wave solution

− i ω t

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).

, then the solitary wave

is Lyapunov stable, with the Lyapunov function given by

is the energy of a solution

, as long as the constant

is chosen sufficiently large.