Special Algorithms for Exponentially Small Phenomena

Abstract

Standard infinite interval spectral methods fail for nonlocal solitary waves. The core is well-approximated, but the wings contain an infinite number of crests and troughs. It is ridiculous to imagine that a finite number of rational Chebyshev functions (or whatever) can faithfully mimic a never-ending oscillation over all of space. To obtain a solution which is uniformly accurate everywhere with an error smaller than O(α) where α is (as always) the amplitude of the wings, one has only two options.

“There are, so to speak, in the mathematical country, precipices and pit-shafts down which it would be possible to fall, but that need not deter us from walking about.” —Lewis F. Richardson, in “How to solve differential equations approx–imately by arithmetic”, Mathematical Gazette, 12, 415–421 (1925); reprinted in his Collected Works.