Existence of periodic traveling wave solutions to the generalized forced Boussinesq equation
type: Item
instance of: scholarly article
title: Existence of periodic traveling wave solutions to the generalized forced Boussinesq equation
zbMATH Open document ID: 0961.35142
DOI: 10.1155/S0161171299226439
author: Kenneth L. Jones
author: Yunkai Chen
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2000-12-03T00:00:00Z
full work available at URL: https://eudml.org/doc/48736
review text: Summary: The generalized forced Boussinesq equation, \[ u_{tt}- u_{xx}+ [f(u)]_{xx}+ u_{xxxx}= h_0, \] and its periodic traveling wave solutions are considered. Using the transform \(z= x-\omega t\), the equation is converted to a nonlinear ordinary differential equation with periodic boundary conditions. An equivalent relation between the ordinary differential equation and a Hammerstein type integral inequation is then established by using the Green's function method. This integral equation generates compact operators in a Banach space of real-valued continuous periodic functions with a given period \(2T\). The Schauder's fixed point theorem is then used to prove the existence of solutions to the integral equation. Therefore, the existence of nonconstant periodic traveling wave solutions to the generalized forced Boussinesq equation is established.
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 35B10
Mathematics Subject Classification ID: 76B15
zbMATH DE Number: 1538456
zbMATH Keywords: nonlinear ordinary differential equation
zbMATH Keywords: Hammerstein type integral inequation
zbMATH Keywords: existence of nonconstant periodic traveling wave solutions
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1538456