 Orbital stability - Wikiwand

# Orbital stability

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

## Formal definition

Formal definition is as follows. Consider the dynamical system

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

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

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

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

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

$i{\frac {\partial }{\partial t))u=-{\frac {\partial ^{2)){\partial x^{2))}u+g\!\left(|u|^{2}\right)u,\qquad u(x,t)\in \mathbb {C} ,\quad x\in \mathbb {R} ,\quad t\in \mathbb {R} ,$ where $g$ is a smooth real-valued function, is orbitally stable if the Vakhitov–Kolokolov stability criterion is satisfied:

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

$Q(u)={\frac {1}{2))\int _{\mathbb {R} }|u(x,t)|^{2}\,dx$ is the charge of the solution $u(x,t)$ , which is conserved in time (at least if the solution $u(x,t)$ is sufficiently smooth).

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