Existence of periodic traveling wave solution to the forced generalized nearly concentric Korteweg-de Vries equation
type: Item
instance of: scholarly article
title: Existence of periodic traveling wave solution to the forced generalized nearly concentric Korteweg-de Vries equation
zbMATH Open document ID: 0961.35143
DOI: 10.1155/S0161171200004336
author: Kenneth L. Jones
author: Xiaogui He
author: Yunkai Chen
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2001-05-28T00:00:00Z
full work available at URL: https://eudml.org/doc/48737
review text: Summary: This paper is concerned with periodic traveling wave solutions of the forced generalized nearly concentric Korteweg-de Vries (KdV) equation in the form of \[ (u_\eta+ u/(2\eta)+ [f(u)]_\xi+ u_{\xi\xi\xi})_\xi+ u_{\theta\theta}/\eta^2= h_0. \] The authors first convert this equation into a forced generalized Kadomtsev-Petviashvili (KP) equation, \[ (u_t+ [f(u)]_x+ u_{xxx})_x+ u_{yy}= h_0, \] and then to a nonlinear ordinary differential equation with periodic boundary conditions. An equivalent relationship between the ordinary differential equation and nonlinear integral equations with symmetric kernels is established by using the Green's function method. The integral representations generate compact operators in a Banach space of real-valued continuous functions. The Schauder's fixed point theorem is then used to prove the existence of nonconstant solutions to the integral equations. Therefore, the existence of periodic traveling wave solutions to the forced generalized KP equation, and hence the nearly concentric KdV equation, is proved.
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 35B10
Mathematics Subject Classification ID: 35B20
Mathematics Subject Classification ID: 76B15
zbMATH DE Number: 1548199
zbMATH Keywords: generalized Kadomtsev-Petviashvili equation
zbMATH Keywords: nonlinear ordinary differential equation
zbMATH Keywords: existence of periodic traveling wave solutions
MaRDI profile type: Publication
Recommended article: Q4340175
Recommended article: Nearly concentric Korteweg-de Vries equation and periodic traveling wave solution
Recommended article: Q3500412
Recommended article: Q3640212
Recommended article: On existence theorems of periodic traveling wave solutions to the two-dimensional korteweg-devries equation
Recommended article: Existence of periodic traveling wave solutions to the generalized forced Boussinesq equation
Recommended article: Anti-periodic traveling wave solutions to a forced two-dimensional generalized Korteweg-de Vries equation
Recommended article: Existence of travelling-wave solutions to a coupled system of Korteweg-de Vries equations
Recommended article: Q4834660
Recommended article: Existence of Periodic Traveling Wave Solutions for a K-P-Boussinesq Type System
description: scientific article; zbMATH DE number 1548199