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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 .
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
where is a smooth real-valued function, is orbitally stable if the Vakhitov–Kolokolov stability criterion is satisfied:
is the charge of the solution , which is conserved in time (at least if the solution is sufficiently smooth).
It was also shown, 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.
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- 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.
- 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.
- 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.
- Richard Jordan & Bruce Turkington (2001). "Statistical equilibrium theories for the nonlinear Schrödinger equation". Advances in Wave Interaction and Turbulence. Contemp. Math. 283. South Hadley, MA. pp. 27–39. doi:10.1090/conm/283/04711. ISBN 9780821827147.
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