Singular perturbation to shock wave solutions of Burgers KdV-equation.
type: Item
instance of: scholarly article
title: Singular perturbation to shock wave solutions of Burgers KdV-equation.
zbMATH Open document ID: 1039.35103
DOI: 10.1016/S0022-247X(02)00493-6
author: Wenjiang J. Fu
author: Jimin Zhang
author: Fuquan Pu
published in: Journal of Mathematical Analysis and Applications
publication date: 2004-01-06T00:00:00Z
review text: The paper considers traveling-wave solutions to the mixed Korteweg-de Vries (KdV)-Burgers equation, in the regime which is close to the dissipative Burgers equation. In the latter case, well-known exact solutions are available, which describe traveling shock waves; from the viewpoint of the corresponding ODE, obtained from the underlying PDE by the traveling-wave substitution, these solutions are heteroclinic trajectories. The authors give a rigorous proof of the fact that shock-wave solutions exist as well in the general case of the mixed KdV-Burgers equation (when they are not available in an exact form), and these solutions smoothly carry over into the exact shock waves of the Burgers equation in the limit when the equation itself tends to the Burgers' form. Additionally, it is proved that, for fixed boundary values, the shock-wave solution (in the case of the general mixed equation) is a single possible solution.
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 76L05
Mathematics Subject Classification ID: 34C37
zbMATH DE Number: 2022060
zbMATH Keywords: Korteweg-de Vries equation
zbMATH Keywords: traveling wave
zbMATH Keywords: heteroclinic trajectory
zbMATH Keywords: mixed KdV-Burgers equation
reviewed by: B. A. Malomed
MaRDI profile type: Publication
cites work: Weak Nonlinear Dispersive Waves: A Discussion Centered Around the Korteweg–De Vries Equation
cites work: An asymptotic solution for the KdVB monotone shock wave and its generalisation
full work available at URL: s0022-247x(02)00493-6
OpenAlex ID: W2059685028
description: scientific article; zbMATH DE number 2022060