In mathematical physics and the theory of partial differential equations, the solitary wave solution of the form u ( x , t ) = e − i ω t ϕ ( 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 ϕ ( x ) {\displaystyle \phi (x)} forever remains in a given small neighborhood of the trajectory of e − i ω t ϕ ( x ) . {\displaystyle e^{-i\omega t}\phi (x).}
Formal definition Formal definition is as follows. Consider the dynamical system
i d u d t = A ( u ) , u ( t ) ∈ X , t ∈ R , {\displaystyle i{\frac {du}{dt}}=A(u),\qquad u(t)\in X,\quad t\in \mathbb {R} ,}
with X {\displaystyle X} a Banach space over C {\displaystyle \mathbb {C} } , and A : X → X {\displaystyle A:X\to X} . We assume that the system is U ( 1 ) {\displaystyle \mathrm {U} (1)} -invariant, so that
A ( e i s u ) = e i s A ( u ) {\displaystyle A(e^{is}u)=e^{is}A(u)} for any u ∈ X {\displaystyle u\in X} and any s ∈ R {\displaystyle s\in \mathbb {R} } . Assume that ω ϕ = A ( ϕ ) {\displaystyle \omega \phi =A(\phi )} , so that u ( t ) = e − i ω t ϕ {\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 e − i ω t ϕ {\displaystyle e^{-i\omega t}\phi } is orbitally stable if for any ϵ > 0 {\displaystyle \epsilon >0} there is δ > 0 {\displaystyle \delta >0} such that for any v 0 ∈ X {\displaystyle v_{0}\in X} with ‖ ϕ − v 0 ‖ X < δ {\displaystyle \Vert \phi -v_{0}\Vert _{X}<\delta } there is a solution v ( t ) {\displaystyle v(t)} defined for all t ≥ 0 {\displaystyle t\geq 0} such that v ( 0 ) = v 0 {\displaystyle v(0)=v_{0}} , and such that this solution satisfies
sup t ≥ 0 inf s ∈ R ‖ v ( t ) − e i s ϕ ‖ X < ϵ . {\displaystyle \sup _{t\geq 0}\inf _{s\in \mathbb {R} }\Vert v(t)-e^{is}\phi \Vert _{X}<\epsilon .}
Example According to , the solitary wave solution e − i ω t ϕ ω ( x ) {\displaystyle e^{-i\omega t}\phi _{\omega }(x)} to the nonlinear Schrödinger equation
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