Exponentially Small Estimates for Separatrix Splittings

Scheurle, J., J. E. marsden, and P. Holmes

Proc. Conf. Beyond all Orders, Birkhäuser, Boston, H. Segur and S. Tanveer, eds.

Abstract:

This paper reviews our previous estimates and gives an example exhibiting a new phenomenon. In problems involving asymptotics beyond all orders in a perturbation parameter $\varepsilon$, it is a common assumption that the quantity being studied (such as a separatrix splitting distance or angle, a solitary wave mismatch, etc.) can be ``estimated'' by an expression of the form a$\varepsilon^{b}_{}$e<sup>-c/$\varepsilon$</sup> as $\varepsilon$$\to$ 0. Here, a, b and c are constants (where b can be negative and c is ``sharp'', often the distance from the real axis to a pole in the complex plane). The main purpose of our example is to show that this assumption can be wrong. The example, which concerns the splitting of separatrices in a rapidly forced system with a heteroclinic orbit shows that even the estimate from above (using the sharp value of c) can be incorrect. We argue that this situation is not isolated or particular, but happens rather generally. We especially note that in situations involving asymptotics beyond all orders, when an estimate of the form a$\varepsilon^{b}_{}$e^{-c/$\varepsilon$} is assumed, it needs to be justified.