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.¹ Consider the dynamical system

${\displaystyle i{\frac {du}{dt}}=A(u),\qquad u(t)\in X,\quad t\in \mathbb {R} ,}$

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

${\displaystyle \sup _{t\geq 0}\inf _{s\in \mathbb {R} }\Vert v(t)-e^{is}\phi \Vert _{X}<\epsilon .}$

According to ² ,³ the solitary wave solution ${\displaystyle e^{-i\omega t}\phi _{\omega }(x)}$ to the nonlinear Schrödinger equation

${\displaystyle i{\frac {\partial }{\partial t}}u=-{\frac {\partial ^{2}}{\partial x^{2}}}u+g\!\left(|u|^{2}\right)u,\qquad u(x,t)\in \mathbb {C} ,\quad x\in \mathbb {R} ,\quad t\in \mathbb {R} ,}$

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

${\displaystyle {\frac {d}{d\omega }}Q(\phi _{\omega })<0,}$

where

${\displaystyle Q(u)={\frac {1}{2}}\int _{\mathbb {R} }|u(x,t)|^{2}\,dx}$

is the charge of the solution ${\displaystyle u(x,t)}$, which is conserved in time (at least if the solution ${\displaystyle u(x,t)}$ is sufficiently smooth).

It was also shown,⁴⁵ that if ${\textstyle {\frac {d}{d\omega }}Q(\omega )<0}$ at a particular value of ${\displaystyle \omega }$, then the solitary wave ${\displaystyle e^{-i\omega t}\phi _{\omega }(x)}$ is Lyapunov stable, with the Lyapunov function given by ${\displaystyle L(u)=E(u)-\omega Q(u)+\Gamma (Q(u)-Q(\phi _{\omega }))^{2}}$, where ${\displaystyle E(u)={\frac {1}{2}}\int _{\mathbb {R} }\left(\left|{\frac {\partial u}{\partial x}}\right|^{2}+G\!\left(|u|^{2}\right)\right)dx}$ is the energy of a solution ${\displaystyle u(x,t)}$, with ${\textstyle G(y)=\int _{0}^{y}g(z)\,dz}$ the antiderivative of ${\displaystyle g}$, as long as the constant ${\displaystyle \Gamma >0}$ is chosen sufficiently large.