In mathematical physics and the theory of partial differential equations, the solitary wave solution of the form 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

Formal definition

Formal definition is as follows. Consider the dynamical system with X a Banach space over \Complex, and A : X \to X. We assume that the system is -invariant, so that for any u\in X and any s\in\R. Assume that, so that is a solution to the dynamical system. We call such solution a solitary wave. We say that the solitary wave is orbitally stable if for any there is \delta > 0 such that for any v_0\in X with there is a solution v(t) defined for all t\ge 0 such that v(0) = v_0, and such that this solution satisfies

Example

According to , the solitary wave solution to the nonlinear Schrödinger equation where g is a smooth real-valued function, is orbitally stable if the Vakhitov–Kolokolov stability criterion is satisfied: where 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 at a particular value of \omega, then the solitary wave is Lyapunov stable, with the Lyapunov function given by, where is the energy of a solution u(x,t), with the antiderivative of g, as long as the constant \Gamma>0 is chosen sufficiently large.