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.

See also

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