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

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

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: 27–39. doi:10.1090/conm/283/04711.