Perturbation Analysis of a Problem of Carrier's

Abstract

In this paper, we analyze the asymptotic behavior of a nonautonomous nonlinear problem proposed by G. F. Carrier. In addition to its historical interest, this problem presents some unusual features; the internal layers, instead of obeying an approximate equal spacing rule as in the famous "spurious solutions" problem, in fact, coalesce. This feature is revealed by unusual matching that incorporates exponentially small and large terms in the matching process. Symmetry notions also play a role in understanding this interesting phenomenon. The asymptotics are compared with full numerical solutions.

Analysis of Carrier's Problem

“…for some constant a as the turning point is approached with the result that the separation between spikes is O( log ) there, in agreement with the analysis in [15] on the two spike solution.…”

“…They showed that the spikes must be symmetrically placed about x = 0, and that the separation between them is O( log ). In view of the rather intricate asymptotic analysis in [15], it is perhaps not surprising that no attempt has been made to analyze the three spike solutions.…”

“…where we have matched with the outer solution by requiring that y in → −1 − √ 2 as |X| → ∞. The constant A (corresponding to a translation in the centre of the spike) is left undetermined in [3], although it is shown in [15] that it must be zero. Since this internal spike solution can be combined with any combination of boundary layers at x = ±1, we have generated another four solutions to (1.1).…”

“…There seems to have been remarkably little work following up on this claim. MacGillivray et al [15] considered the solutions with two spikes in detail. They showed that the spikes must be symmetrically placed about x = 0, and that the separation between them is O( log ).…”

On the Number of Solutions to Carrier's Problem

“…Because the equation in is nonautonomous, the difficulty with this problem is even greater. There are other studies on this problem in the literature; see, for instance, Kath [8] and MacGillivary et al [11]. In the latter reference, the authors also showed that the spikes will cluster in the neighborhood of x = 0.…”