In mathematical physics or theory of partial differential equations, the solitary wave solution of the form u ( x , t ) = e − i ω t ϕ ( x ) is said to be orbitally stable if any solution with the initial data sufficiently close to ϕ ( x ) forever remains in a given small neighborhood of the trajectory of e − i ω t ϕ ( x ) .
Formal definition is as follows. Let us consider the dynamical system
i d u d t = A ( u ) , u ( t ) ∈ X , t ∈ R , with X a Banach space over C , and A : X → X . We assume that the system is U ( 1 ) -invariant, so that A ( e i s u ) = e i s A ( u ) for any u ∈ X and any s ∈ R .
Assume that ω ϕ = A ( ϕ ) , so that u ( t ) = e − i ω t ϕ 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 is orbitally stable if for any ϵ > 0 there is δ > 0 such that for any v 0 ∈ X with ∥ ϕ − v 0 ∥ X < δ there is a solution v ( t ) defined for all t ≥ 0 such that v ( 0 ) = v 0 , and such that this solution satisfies
sup t ≥ 0 inf s ∈ R ∥ v ( t ) − e i s ϕ ∥ X < ϵ . According to , the solitary wave solution e − i ω t ϕ ω ( x ) to the nonlinear Schrödinger equation
i ∂ ∂ t u = − ∂ 2 ∂ x 2 u + g ( | u | 2 ) u , u ( x , t ) ∈ C , x ∈ R , t ∈ R , where g is a smooth real-valued function, is orbitally stable if the Vakhitov–Kolokolov stability criterion is satisfied:
d d ω Q ( ϕ ω ) < 0 , where
Q ( u ) = 1 2 ∫ R | u ( x , t ) | 2 d x 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 d d ω Q ( ω ) < 0 at a particular value of ω , then the solitary wave e − i ω t ϕ ω ( x ) is Lyapunov stable, with the Lyapunov function given by L ( u ) = E ( u ) − ω Q ( u ) + Γ ( Q ( u ) − Q ( ϕ ω ) ) 2 , where E ( u ) = 1 2 ∫ R ( | ∂ u ∂ x | 2 + G ( | u | 2 ) ) d x is the energy of a solution u ( x , t ) , with G ( y ) = ∫ 0 y g ( z ) d z the antiderivative of g , as long as the constant Γ > 0 is chosen sufficiently large.
