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,^[4][5] 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.

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