Orbital stability

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

Formal definition

Formal definition is as follows.[1] Let us consider the dynamical system

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

with X\, a Banach space over \C\,, and A\,:X\to X. We assume that the system is \mathrm{U}(1)\,, so that A(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 t}\phi\, is a solution to the dynamical system. We call such solution a solitary wave.

We say that the solitary wave e^{-i\omega t}\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 0}\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 t}\phi_\omega(x)\, to the nonlinear Schrödinger equation

i\frac{\partial}{\partial t}u=-\frac{\partial^2}{\partial x\,^2}u+g(|u|^2)u, \qquad u(x,t)\in\C,\quad x\in\R,\quad t\in\R,

where g\, is a smooth real-valued function, is orbitally stable if the Vakhitov–Kolokolov stability criterion is satisfied:

\frac{d}{d\omega}Q(\phi_\omega)<0,

where

Q(u)=\frac{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 \frac{d}{d\omega}Q(\omega)<0 at a particular value of \omega\,, then the solitary wave e^{-i\omega t}\phi_\omega(x)\, is Lyapunov stable, with the Lyapunov function given by L(u)=E(u)-\omega Q(u)+\Gamma(Q(u)-Q(\phi_\omega))^2\,, where E(u)=\frac{1}{2}\int_{\R}\left(|\frac{\partial u}{\partial x}|^2+G(|u|^2)\right)\,dx is the energy of a solution u(x,t)\,, with G(y)=\int_0^y g(z)\,dz the antiderivative of g\,, as long as the constant \Gamma>0\, is chosen sufficiently large.

See also

References

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