Orbital stability

In mathematical physics and the theory of partial differential equations, the solitary wave solution of the form ${\displaystyle u(x,t)=e^{-i\omega t}\phi (x)}${\displaystyle 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 ${\displaystyle \phi (x)}${\displaystyle \phi (x)} forever remains in a given small neighborhood of the trajectory of ${\displaystyle e^{-i\omega t}\phi (x).}${\displaystyle e^{-i\omega t}\phi (x).}

Formal definition is as follows.^[1] Consider the dynamical system

${\displaystyle i{\frac {du}{dt}}=A(u),\qquad u(t)\in X,\quad t\in \mathbb {R} ,}${\displaystyle i{\frac {du}{dt}}=A(u),\qquad u(t)\in X,\quad t\in \mathbb {R} ,}

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

Assume that ${\displaystyle \omega \phi =A(\phi )}${\displaystyle \omega \phi =A(\phi )}, so that ${\displaystyle u(t)=e^{-i\omega t}\phi }${\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 ${\displaystyle e^{-i\omega t}\phi }${\displaystyle e^{-i\omega t}\phi } is orbitally stable if for any ${\displaystyle \epsilon >0}${\displaystyle \epsilon >0} there is ${\displaystyle \delta >0}${\displaystyle \delta >0} such that for any ${\displaystyle v_{0}\in X}${\displaystyle v_{0}\in X} with ${\displaystyle \Vert \phi -v_{0}\Vert _{X}<\delta }${\displaystyle \Vert \phi -v_{0}\Vert _{X}<\delta } there is a solution ${\displaystyle v(t)}${\displaystyle v(t)} defined for all ${\displaystyle t\geq 0}${\displaystyle t\geq 0} such that ${\displaystyle v(0)=v_{0}}${\displaystyle v(0)=v_{0}}, and such that this solution satisfies

${\displaystyle \sup _{t\geq 0}\inf _{s\in \mathbb {R} }\Vert v(t)-e^{is}\phi \Vert _{X}<\epsilon .}${\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 ${\displaystyle e^{-i\omega t}\phi _{\omega }(x)}${\displaystyle e^{-i\omega t}\phi _{\omega }(x)} to the nonlinear Schrödinger equation

${\displaystyle i{\frac {\partial }{\partial t}}u=-{\frac {\partial ^{2}}{\partial x^{2}}}u+g\!\left(|u|^{2}\right)u,\qquad u(x,t)\in \mathbb {C} ,\quad x\in \mathbb {R} ,\quad t\in \mathbb {R} ,}${\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 ${\displaystyle g}${\displaystyle g} is a smooth real-valued function, is orbitally stable if the Vakhitov–Kolokolov stability criterion is satisfied:

${\displaystyle {\frac {d}{d\omega }}Q(\phi _{\omega })<0,}${\displaystyle {\frac {d}{d\omega }}Q(\phi _{\omega })<0,}

where

${\displaystyle Q(u)={\frac {1}{2}}\int _{\mathbb {R} }|u(x,t)|^{2},円dx}${\displaystyle Q(u)={\frac {1}{2}}\int _{\mathbb {R} }|u(x,t)|^{2},円dx}

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

It was also shown,^{[4] [5]} that if ${\textstyle {\frac {d}{d\omega }}Q(\omega )<0}${\textstyle {\frac {d}{d\omega }}Q(\omega )<0} at a particular value of ${\displaystyle \omega }${\displaystyle \omega }, then the solitary wave ${\displaystyle e^{-i\omega t}\phi _{\omega }(x)}${\displaystyle e^{-i\omega t}\phi _{\omega }(x)} is Lyapunov stable, with the Lyapunov function given by ${\displaystyle L(u)=E(u)-\omega Q(u)+\Gamma (Q(u)-Q(\phi _{\omega }))^{2}}${\displaystyle L(u)=E(u)-\omega Q(u)+\Gamma (Q(u)-Q(\phi _{\omega }))^{2}}, where ${\displaystyle E(u)={\frac {1}{2}}\int _{\mathbb {R} }\left(\left|{\frac {\partial u}{\partial x}}\right|^{2}+G\!\left(|u|^{2}\right)\right)dx}${\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 ${\displaystyle u(x,t)}${\displaystyle u(x,t)}, with ${\textstyle G(y)=\int _{0}^{y}g(z),円dz}${\textstyle G(y)=\int _{0}^{y}g(z),円dz} the antiderivative of ${\displaystyle g}${\displaystyle g}, as long as the constant ${\displaystyle \Gamma >0}${\displaystyle \Gamma >0} is chosen sufficiently large.

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

• Stability theory
• Solitons

• CS1 maint: location missing publisher
• Articles with short description
• Short description is different from Wikidata