Orbital stability

In mathematical physics and the theory of partial differential equations, the solitary wave solution of the form ${\displaystyle u(x,t)=e^{-i\omega t}\phi (x)\,}$ is said to be orbitally stable if any solution with the initial data sufficiently close to ${\displaystyle \phi (x)\,}$ forever remains in a given small neighborhood of the trajectory of ${\displaystyle e^{-i\omega t}\phi (x)\,}$.

Formal definition

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

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

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

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

We say that the solitary wave ${\displaystyle e^{-i\omega t}\phi \,}$ is orbitally stable if for any ${\displaystyle \epsilon >0\,}$ there is ${\displaystyle \delta >0\,}$ such that for any ${\displaystyle v_{0}\in X}$ with ${\displaystyle \Vert \phi -v_{0}\Vert _{X}<\delta \,}$ there is a solution ${\displaystyle v(t)\,}$ defined for all ${\displaystyle t\geq 0}$ such that ${\displaystyle v(0)=v_{0}\,}$, and such that this solution satisfies

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

Example

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

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

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

${\displaystyle {\frac {d}{d\omega }}Q(\phi _{\omega })<0,}$

where

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

is the charge of the solution ${\displaystyle u(x,t)\,}$, which is conserved in time (at least if the solution ${\displaystyle u(x,t)\,}$ is sufficiently smooth).

It was also shown,[4][5] that if ${\displaystyle {\frac {d}{d\omega }}Q(\omega )<0}$ at a particular value of ${\displaystyle \omega \,}$, then the solitary wave ${\displaystyle e^{-i\omega t}\phi _{\omega }(x)\,}$ is Lyapunov stable, with the Lyapunov function given by ${\displaystyle L(u)=E(u)-\omega Q(u)+\Gamma (Q(u)-Q(\phi _{\omega }))^{2}\,}$, where ${\displaystyle E(u)={\frac {1}{2}}\int _{\mathbb {R} }\left(|{\frac {\partial u}{\partial x}}|^{2}+G(|u|^{2})\right)\,dx}$ is the energy of a solution ${\displaystyle u(x,t)\,}$, with ${\displaystyle G(y)=\int _{0}^{y}g(z)\,dz}$ the antiderivative of ${\displaystyle g\,}$, as long as the constant ${\displaystyle \Gamma >0\,}$ 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. Cite journal requires |journal= (help)