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\R, $$

with $$X$$ a Banach space over $$\Complex$$, 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 $$s\in\R$$.

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 $$t\ge 0$$ such that $$v(0) = v_0$$, and such that this solution satisfies


 * $$\sup_{t\ge 0} \inf_{s\in\R} \Vert v(t) - e^{is} \phi \Vert_X < \epsilon.$$

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\Complex,\quad x\in\R,\quad t\in\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


 * $$Q(u) = \frac{1}{2} \int_{\R} |u(x,t)|^2 \, dx$$

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 $$E(u) = \frac{1}{2} \int_{\R} \left(\left|\frac{\partial u}{\partial x}\right|^2 + G\!\left(|u|^2\right)\right) dx$$ is the energy of a solution $$u(x,t)$$, with $G(y) = \int_0^y g(z)\,dz$ the antiderivative of $$g$$, as long as the constant $$\Gamma>0$$ is chosen sufficiently large.