Orbital Instability

In mathematical physics and the theory of partial differential equations, the solitary wave solution of the form $u(x,t)=e^\phi(x)$ 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 $e^\phi(x).$

Formal definition

Formal definition is as follows. Consider the dynamical system

:$i\frac=A(u), \qquad u(t)\in X, \quad t\in\R,$

with $X$ a Banach space over $\Complex$, and $A : X \to X$. We assume that the system is $\mathrm(1)$-invariant,
so that
$A(e^u) = e^A(u)$ for any $u\in X$ and any $s\in\R$.

Assume that $\omega \phi=A(\phi)$, so that $u(t)=e^\phi$ is a solution to the dynamical system.
We call such solution a solitary wave.

We say that the solitary wave $e^\phi$ 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

:$\sup_ \inf_ \Vert v(t) - e^ \phi \Vert_X < \epsilon.$

Example

According to
,
the solitary wave solution $e^\phi_\omega(x)$ to the nonlinear Schrödinger equation
:$i\frac u = -\frac u+g\!\left(, u, ^2\right)u, \qquad u(x,t)\in\Complex,\quad x\in\R,\quad t\in\R,$
where $g$ is a smooth real-valued function, is orbitally stable if the Vakhitov–Kolokolov stability criterion is satisfied:

: $\fracQ(\phi_\omega) < 0,$

where

: $Q(u) = \frac \int_ , u(x,t), ^2 \, dx$

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 $\fracQ(\omega) < 0$ at a particular value of $\omega$, then the solitary wave
$e^\phi_\omega(x)$ is Lyapunov stable, with the Lyapunov function
given by $L(u) = E(u) - \omega Q(u) + \Gamma(Q(u)-Q(\phi_\omega))^2$, where $E(u) = \frac \int_ \left(\left, \frac{\partial x}\^2 + G\!\left(, u, ^2\right)\right) dx$ 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.

See also

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

References

Stability theory
Solitons