Wave number shocks for the leading tail of Korteweg-de Vries solitary waves in slowly varying media
type: Item
instance of: scholarly article
title: Wave number shocks for the leading tail of Korteweg-de Vries solitary waves in slowly varying media
zbMATH Open document ID: 0731.35088
DOI: 10.1016/0165-2125(90)90026-Z
author: Darrell E. Allgaier
author: Richard Haberman
published in: Wave Motion
publication date: 1990-01-01T00:00:00Z
review text: The leading tail for slowly varying solitary waves for the perturbed Korteweg-de Vries (KdV) equation is analyzed. The path of the core of the solitary wave is obtained and shown to provide a moving boundary for the leading tail. The leading tail is predicted to be triple valued within a penumbral caustic (envelope of characteristics) caused by the initial acceleration of the core. A rescaling in the neighborhood of the singularity shows that the solution there satisfies the diffusion equation. The solution involves an incomplete Airy-type exponential integral, where critical points (significant for Laplace's asymptotic method) satisfy the structure of the penumbral caustic. A wave number shock develops, which separates two different solitary wave tails, one due to the moving core and the other due to the initial condition. The shock velocity is that predicted from conservation of waves.
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 35L67
zbMATH DE Number: 4208693
zbMATH Keywords: perturbed Korteweg-de Vries equation
zbMATH Keywords: solitary waves
zbMATH Keywords: leading tail
zbMATH Keywords: shock velocity
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4208693