Nonexistence of small periodic traveling wave solutions to the power Kadomtsev-Petviashvili equation
type: Item
instance of: scholarly article
title: Nonexistence of small periodic traveling wave solutions to the power Kadomtsev-Petviashvili equation
zbMATH Open document ID: 0891.34021
author: Yunkai Chen
author: Xiaogui He
published in: Differential Equations and Dynamical Systems
publication date: 1997-12-18T00:00:00Z
review text: This paper is concerned with the nonexistence of small periodic traveling wave solutions to the power Kadomtsev-Petviashvili equation of the form \[ [u_t+\gamma u^n u_x+ \alpha u_{xxx}]_x+ \beta u_{yy}= 0. \] This equation is converted to a periodic boundary value problem for a nonlinear fourth-order ordinary differential equation, which is shown to be equivalent to a nonlinear integral equation with kernel of Green function type. The authors define a compact operator \(A\) on the Banach space \(C_{2T}\) of real-valued continuous periodic functions with a given period \(2T\). They prove that for any small \(T>0\), there exists an \(r>0\) such that \(A\) has no nontrivial fixed point in \(B(0,r)\subset C_{2T}\).
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 45G10
zbMATH DE Number: 1098671
zbMATH Keywords: nonexistence
zbMATH Keywords: periodic traveling wave solutions
zbMATH Keywords: power Kadomtsev-Petviashvili equation
zbMATH Keywords: periodic boundary value problem
zbMATH Keywords: Green function
reviewed by: Abdelkader Boucherif
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1098671