Orbital stability

Short description: Solution to a partial differential equation which remains close to the initial data

In mathematical physics and the theory of partial differential equations, the solitary wave solution of the form [math]\displaystyle{ u(x,t)=e^{-i\omega t}\phi(x) }[/math] is said to be orbitally stable if any solution with the initial data sufficiently close to [math]\displaystyle{ \phi(x) }[/math] forever remains in a given small neighborhood of the trajectory of [math]\displaystyle{ e^{-i\omega t}\phi(x). }[/math]

Formal definition

Formal definition is as follows. Consider the dynamical system

[math]\displaystyle{ i\frac{du}{dt}=A(u), \qquad u(t)\in X, \quad t\in\R, }[/math]

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

Assume that [math]\displaystyle{ \omega \phi=A(\phi) }[/math], so that [math]\displaystyle{ u(t)=e^{-i\omega t}\phi }[/math] is a solution to the dynamical system. We call such solution a solitary wave.

We say that the solitary wave [math]\displaystyle{ e^{-i\omega t}\phi }[/math] is orbitally stable if for any [math]\displaystyle{ \epsilon \gt 0 }[/math] there is [math]\displaystyle{ \delta \gt 0 }[/math] such that for any [math]\displaystyle{ v_0\in X }[/math] with [math]\displaystyle{ \Vert \phi-v_0\Vert_X \lt \delta }[/math] there is a solution [math]\displaystyle{ v(t) }[/math] defined for all [math]\displaystyle{ t\ge 0 }[/math] such that [math]\displaystyle{ v(0) = v_0 }[/math], and such that this solution satisfies

[math]\displaystyle{ \sup_{t\ge 0} \inf_{s\in\R} \Vert v(t) - e^{is} \phi \Vert_X \lt \epsilon. }[/math]

Example

According to , the solitary wave solution [math]\displaystyle{ e^{-i\omega t}\phi_\omega(x) }[/math] to the nonlinear Schrödinger equation

[math]\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\Complex,\quad x\in\R,\quad t\in\R, }[/math]

where [math]\displaystyle{ g }[/math] is a smooth real-valued function, is orbitally stable if the Vakhitov–Kolokolov stability criterion is satisfied:

[math]\displaystyle{ \frac{d}{d\omega}Q(\phi_\omega) \lt 0, }[/math]

where

[math]\displaystyle{ Q(u) = \frac{1}{2} \int_{\R} |u(x,t)|^2 \, dx }[/math]

is the charge of the solution [math]\displaystyle{ u(x,t) }[/math], which is conserved in time (at least if the solution [math]\displaystyle{ u(x,t) }[/math] is sufficiently smooth).

It was also shown, that if [math]\displaystyle{ \frac{d}{d\omega}Q(\omega) \lt 0 }[/math] at a particular value of [math]\displaystyle{ \omega }[/math], then the solitary wave [math]\displaystyle{ e^{-i\omega t}\phi_\omega(x) }[/math] is Lyapunov stable, with the Lyapunov function given by [math]\displaystyle{ L(u) = E(u) - \omega Q(u) + \Gamma(Q(u)-Q(\phi_\omega))^2 }[/math], where [math]\displaystyle{ E(u) = \frac{1}{2} \int_{\R} \left(\left|\frac{\partial u}{\partial x}\right|^2 + G\!\left(|u|^2\right)\right) dx }[/math] is the energy of a solution [math]\displaystyle{ u(x,t) }[/math], with [math]\displaystyle{ G(y) = \int_0^y g(z)\,dz }[/math] the antiderivative of [math]\displaystyle{ g }[/math], as long as the constant [math]\displaystyle{ \Gamma\gt 0 }[/math] is chosen sufficiently large.

See also

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