Orbital stability

Example

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

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

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

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

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, that if $\frac{d}{d\omega }Q(\omega )<0$ at a particular value of ${\displaystyle \omega }$, then the solitary wave ${\displaystyle e^{-i\omega t}{\varphi }_{\omega }(x)}$ is Lyapunov stable, with the Lyapunov function given by ${\displaystyle L(u)=E(u)-\omega Q(u)+\mathrm{\Gamma }(Q(u)-Q({\varphi }_{\omega }){)}^2}$, where ${\displaystyle E(u)=\frac{1}{2}{\int }_{\mathbb{R}}\left({\left|\frac{\mathrm{\partial }u}{\mathrm{\partial }x}\right|}^2+G\left(|u{|}^2\right)\right)dx}$ is the energy of a solution ${\displaystyle u(x,t)}$, with $G(y)={\int }_0^{y}g(z)dz$ the antiderivative of ${\displaystyle g}$, as long as the constant ${\displaystyle \mathrm{\Gamma }>0}$ is chosen sufficiently large.