Orbital stability

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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[edit]

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[edit]

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[edit]

References[edit]

  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.