Ε-net (computational geometry)

In mathematical physics or 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. Let us consider the dynamical system

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

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

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

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 ${\frac {d}{d\omega }}Q(\omega )<0$ 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 $E(u)={\frac {1}{2}}\int _{\mathbb {R} }\left(|{\frac {\partial u}{\partial x}}|^{2}+G(|u|^{2})\right)\,dx$ is the energy of a solution $u(x,t)\,$, with $G(y)=\int _{0}^{y}g(z)\,dz$ the antiderivative of $g\,$, as long as the constant $\Gamma >0\,$ is chosen sufficiently large.

See also

• Stability theory
  • Asymptotic stability
  • Linear stability
  • Lyapunov stability
  • Vakhitov−Kolokolov stability criterion