On the exact solutions of the intermediate long-wave equation
type: Item
instance of: scholarly article
title: On the exact solutions of the intermediate long-wave equation
zbMATH Open document ID: 0937.35142
published in: Differential and Integral Equations
publication date: 2000-06-07T00:00:00Z
review text: The long-wave notion was introduced by R. I. Joseph in 1977. It is a nonlinear dispersive wave at the interface between two fluids of different densities at rest in a long, horizontal channel in the \(x\)-direction. The pseudo-differential operator equation for the unknown function \(\eta (x,t)\) is \[ \eta_t+2 \eta\eta_x- (N_H\eta)_x+ (1/H)\eta_x=0, \] where \(H>0\) is a constant and \(N_H\) is given by \(N_H\widehat\eta (k)=(k \text{coth} H)\widehat \eta(k)\) (the circumflex sign denotes the Fourier transform). The authors, based on results by R. I. Joseph (loc. cit.), study the equation \[ (N+\gamma) \Phi= \Phi^2,\;\gamma>-1 \] for which they prove the main theorem of the paper concerning existence and properties of the solution.
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 76B15
Mathematics Subject Classification ID: 47G30
Mathematics Subject Classification ID: 42A38
zbMATH DE Number: 597480
zbMATH Keywords: long-wave equation
zbMATH Keywords: dispersive waves
zbMATH Keywords: solitary waves
zbMATH Keywords: existence
zbMATH Keywords: properties of the solution
author: John Albert
author: John F. Toland
reviewed by: Vasile Ionescu
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 597480