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

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

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

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 $t\geq 0$ such that $v(0)=v_{0}\,$, and such that this solution satisfies

$\sup _{{t\geq 0}}\inf _{{s\in \mathbb{R} }}\Vert v(t)-e^{{is}}\phi \Vert _{X}<\epsilon .$

Example

According to ^[2] ,^[3] the solitary wave solution $e^{{-i\omega t}}\phi _{\omega }(x)\,$ to the nonlinear Schrödinger equation

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

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

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

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 $E(u)={\frac {1}{2}}\int _{{\mathbb{R} }}\left(|{\frac {\partial u}{\partial x}}|^{2}+G(|u|^{2})\right)\,dx$ 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 & 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 & 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 & 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 & Bruce Turkington (2001). "Statistical equilibrium theories for the nonlinear Schrödinger equation". Contemp. Math. 283: 27–39. doi:10.1090/conm/283/04711.