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

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Formal definition is as follows.[1] 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

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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

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References

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  1. ^ Manoussos Grillakis; Jalal Shatah & Walter Strauss (1990). "Stability theory of solitary waves in the presence of symmetry". J. Funct. Anal. 94 (2): 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. S2CID 120472894.
  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. S2CID 120894859.
  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". Advances in Wave Interaction and Turbulence. Contemp. Math. Vol. 283. South Hadley, MA. pp. 27–39. doi:10.1090/conm/283/04711. ISBN 9780821827147.{{cite book}}: CS1 maint: location missing publisher (link)