Orbital stability explained

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

u(x,t) = e ^{-i\omega} \phi(x) 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 e ^{-i\omega} \phi(x).

Formal definition

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

i (du)/(dt) = A(u), u(t)\in X, t\in\R,

with

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

Assume that

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

We say that the solitary wave

e ^{-i\omega} \phi is orbitally stable if for any \epsilon > 0 there is \delta > 0 such that for any v _{0\in} X with \Vert \phi - v _{0\Vert} _{X} < \delta there is a solution v(t) defined for all t\ge 0 such that v( 0 ) = v _{0} , and such that this solution satisfies\sup _{t\ge} inf _{s\in\R} \Vert v(t) - e ^{is} \phi \Vert _{X} < \epsilon.

Example

According to ^[2],^[3] the solitary wave solution

e ^{-i\omega} \phi _{\omega(x)} to the nonlinear Schrödinger equationi (\partial)/(\partial t) u =- (\partial ^{2})/(\partial x ^{2}) u + g\left(|u| ^{2\right)u, u(x,t)\in\Complex,} x\in\R, t\in\R, where g is a smooth real-valued function, is orbitally stable if the Vakhitov - Kolokolov stability criterion is satisfied:(d)/(d\omega) Q(\phi _{\omega)} < 0 ,

where

Q(u) = (1)/(2) \int _{\R} |u(x,t)| ^{2} dx

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,^{[4] [5]} that if at a particular value of

\omega , then the solitary wavee ^{-i\omega} \phi _{\omega(x)} is Lyapunov stable, with the Lyapunov functiongiven by L(u) = E(u) - \omega Q(u) + \Gamma(Q(u) - Q(\phi ^{2} _{\omega))} , where E(u) = (1)/(2) \int _{\R} \left(\left| (\partial u)/(\partial x) \right| ^{2} + G\left(|u| ^{2\right)\right)} dx is the energy of a solution u(x,t) , with $G(y)\; =\; \backslash int\_0^y\; g(z)\backslash ,dz$ the antiderivative of g , as long as the constant \Gamma> 0 is chosen sufficiently large.

See also

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

References

1. Manoussos Grillakis . Jalal Shatah . Walter Strauss . amp . Stability theory of solitary waves in the presence of symmetry. J. Funct. Anal.. 94. 1990. 2 . 308–348. 10.1016/0022-1236(90)90016-E . free.
2. T. Cazenave . P.-L. Lions. amp . Orbital stability of standing waves for some nonlinear Schrödinger equations. Comm. Math. Phys.. 85. 1982. 4. 549–561. 1982CMaPh..85..549C. 10.1007/BF01403504 . 120472894.
3. Jerry Bona . Panagiotis Souganidis . Walter Strauss . amp . Stability and instability of solitary waves of Korteweg-de Vries type. Proceedings of the Royal Society A. 411. 1987. 1841. 395–412. 10.1098/rspa.1987.0073. 1987RSPSA.411..395B . 120894859 .
4. Michael I. Weinstein . Lyapunov stability of ground states of nonlinear dispersive evolution equations . Comm. Pure Appl. Math. . 39 . 1986 . 1 . 51–67 . 10.1002/cpa.3160390103.
5. Book: Richard Jordan . Bruce Turkington . amp . Statistical equilibrium theories for the nonlinear Schrödinger equation . Advances in Wave Interaction and Turbulence . South Hadley, MA . Contemp. Math. . 283 . 27–39 . 2001 . 10.1090/conm/283/04711. 9780821827147 .