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词条 Orbital stability
释义

  1. Formal definition

  2. Example

  3. See also

  4. References

In mathematical physics and the theory of partial differential equations, the solitary wave solution of the form is said to be orbitally stable if any solution with the initial data sufficiently close to forever remains in a given small neighborhood of the trajectory of .

Formal definition

Formal definition is as follows.[1]

Let us consider the dynamical system

with a Banach space over ,

and .

We assume that the system is

-invariant,

so that

for any

and any .

Assume that ,

so that is a solution to the dynamical system.

We call such solution a solitary wave.

We say that the solitary wave

is orbitally stable if for any there is

such that for any with

there is a solution defined for all

such that ,

and such that this solution satisfies

Example

According to [2]

,[3]

the solitary wave solution

to the nonlinear Schrödinger equation

where is a smooth real-valued function,

is orbitally stable if the Vakhitov–Kolokolov stability criterion is satisfied:

where

is the charge of the solution ,

which is conserved in time (at least if the solution

is sufficiently smooth).

It was also shown,[4][5]

that if at a particular value of ,

then the solitary wave

is Lyapunov stable, with the Lyapunov function

given by ,

where

is the energy of a solution ,

with the antiderivative of ,

as long as the constant

is chosen sufficiently large.

See also

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

References

1. ^{{cite journal|author1=Manoussos Grillakis |author2=Jalal Shatah |author3=Walter Strauss |last-author-amp=yes |title=Stability theory of solitary waves in the presence of symmetry|journal=J. Funct. Anal.|volume=94|year=1990|pages=308–348|doi=10.1016/0022-1236(90)90016-E }}
2. ^{{ cite journal|author1=T. Cazenave |author2=P.-L. Lions |lastauthoramp=yes |title= Orbital stability of standing waves for some nonlinear Schrödinger equations| journal = Comm. Math. Phys.| volume = 85| year = 1982|number = 4|pages=549–561|url=http://projecteuclid.org/getRecord?id=euclid.cmp/1103921547|bibcode=1982CMaPh..85..549C|doi = 10.1007/BF01403504 }}
3. ^{{cite journal|author1=Jerry Bona |author2=Panagiotis Souganidis |author3=Walter Strauss |last-author-amp=yes |title=Stability and instability of solitary waves of Korteweg-de Vries type|journal=Proceedings of the Royal Society A|volume=411|year=1987|issue=1841|pages=395–412|doi=10.1098/rspa.1987.0073|bibcode=1987RSPSA.411..395B }}
4. ^{{cite journal |author = Michael I. Weinstein |title = Lyapunov stability of ground states of nonlinear dispersive evolution equations |journal = Comm. Pure Appl. Math. |volume = 39 |year = 1986 |number = 1 |pages = 51–67 |doi = 10.1002/cpa.3160390103}}
5. ^{{ cite journal |author1=Richard Jordan |author2=Bruce Turkington |lastauthoramp=yes |title = Statistical equilibrium theories for the nonlinear Schrödinger equation |booktitle = Advances in wave interaction and turbulence (South Hadley, MA, 2000) |series = Contemp. Math. |volume = 283 |pages= 27–39 |year = 2001 |doi = 10.1090/conm/283/04711}}

2 : Stability theory|Solitons
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