Exponential Asymptotics and Stokes Line Smoothing for Generalized Solitary Waves

TL;DR

This paper presents an alternative method to analyze generalized solitary waves in the fifth-order Korteweg-de Vries equation, focusing on optimal truncation and Stokes line smoothing to better understand the Stokes Phenomenon.

Contribution

It introduces a new approach for deriving generalized solitary waves by combining optimal truncation with Stokes line smoothing, enhancing understanding of exponentially small terms.

Findings

Explicit view of the switching-on mechanism of exponentially small terms

Improved understanding of the Stokes Phenomenon in solitary waves

Method applicable to higher-order nonlinear wave equations

Abstract

In a companion paper, Grimshaw (Asymptotic Methods in Fluid Mechanics, 2010, pp. 71-120) has demonstrated how techniques of Borel summation can be used to elucidate the exponentially small terms that lie hidden beyond all orders of a divergent asymptotic expansion. Here, we provide an alternative derivation of the generalized solitary waves of the fifth-order Korteweg-de Vries equation. We will first optimally truncate the asymptotic series, and then smooth the Stokes line. Our method provides an explicit view of the switching-on mechanism, and thus increased understanding of the Stokes Phenomenon.