Orbital stability

In mathematical physics and the theory of partial differential equations, the solitary wave solution of the form ${\displaystyle u(x,t)=e^{-i\omega t}\phi (x)}$ is said to be orbitally stable if any solution with the initial data sufficiently close to ${\displaystyle \phi (x)}$ forever remains in a given small neighborhood of the trajectory of ${\displaystyle e^{-i\omega t}\phi (x).}$

Formal definition is as follows. Consider the dynamical system

with ${\displaystyle X}$ a Banach space over ${\displaystyle \mathbb {C} }$, and ${\displaystyle A:X\to X}$. We assume that the system is ${\displaystyle \mathrm {U} (1)}$-invariant, so that ${\displaystyle A(e^{is}u)=e^{is}A(u)}$ for any ${\displaystyle u\in X}$ and any ${\displaystyle s\in \mathbb {R} }$.

Assume that ${\displaystyle \omega \phi =A(\phi )}$, so that ${\displaystyle u(t)=e^{-i\omega t}\phi }$ is a solution to the dynamical system. We call such solution a solitary wave.

We say that the solitary wave ${\displaystyle e^{-i\omega t}\phi }$ is orbitally stable if for any ${\displaystyle \epsilon >0}$ there is ${\displaystyle \delta >0}$ such that for any ${\displaystyle v_{0}\in X}$ with ${\displaystyle \Vert \phi -v_{0}\Vert _{X}<\delta }$ there is a solution ${\displaystyle v(t)}$ defined for all ${\displaystyle t\geq 0}$ such that ${\displaystyle v(0)=v_{0}}$, and such that this solution satisfies

According to , the solitary wave solution ${\displaystyle e^{-i\omega t}\phi _{\omega }(x)}$ to the nonlinear Schrödinger equation

where ${\displaystyle g}$ is a smooth real-valued function, is orbitally stable if the Vakhitov–Kolokolov stability criterion is satisfied:

where