Abstract
Exponential smallness is not a property of solitary waves alone. Rather, exponential dependence on the reciprocal of a small parameter is common in physics, chemistry and engineering (Chapter 17). In this chapter, we shall briefly describe two phenomena which, although not solitons themselves, have very similar mathematics; the analogies to nonlocal solitons are much deeper than mere exponential dependence on 1/∈. An interesting twist is that both are nonlocal in time rather than in a spatial coordinate.
“The magnitude of the splitting of separatrices is exponentially small for small ∈; therefore it is easy to overlook the phenomenon of splitting in calculations in one or another scheme of “perturbation theory”. However this phenomenon is very important in fundamental questions. For example, its existence immedi–ately implies the divergence of the series in numerous versions of perturbation theory (since if the series converged, there would be no splitting).
In general, the divergence of series in perturbation theory (while a good approximation is given by a few initial terms) is usually related to the fact that we are looking for an object which does not exist. If we try to fit a phenomenon to a scheme which actually contradicts the essential features of the phenomenon, then it is not surprising that our series diverge.” — V.I. Arnold, in Mathematical Methods of Classical Mechanics, Springer-Verlag, pgs. 395-396 (1978).
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© 1998 Springer Science+Business Media Dordrecht
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Boyd, J.P. (1998). Temporal Analogues: Separatrix Splitting & the Slow Manifold. In: Weakly Nonlocal Solitary Waves and Beyond-All-Orders Asymptotics. Mathematics and Its Applications, vol 442. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-5825-5_14
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DOI: https://doi.org/10.1007/978-1-4615-5825-5_14
Publisher Name: Springer, Boston, MA
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