Strong instability of solitary-wave solutions to a Kadomtsev-Petviashvili equation in three dimensions
type: Item
instance of: scholarly article
title: Strong instability of solitary-wave solutions to a Kadomtsev-Petviashvili equation in three dimensions
zbMATH Open document ID: 1061.35115
author: Yue Liu
published in: Journal of Differential Equations
publication date: 2002-09-08T00:00:00Z
review text: The author investigates unstable solitary-wave solutions of the generalised Kadomtsev-Petviash\-vili (KP) equation \[ (u_t+u^pu_x+u_{xxx})_x=u_{yy}+u_{zz},\quad (x,y,z)\in{\mathbb R}^2,\;\;t\geq 0, \] with \(p\geq 1.\) Here, the solitary waves are travelling waves of the form \(\phi(x-ct,y,z)\) with \(u\rightarrow 0\) as \(x^2+y^2+z^2\rightarrow \infty\), propagating in the \(x\)-direction. The paper extends the results of a previous work by \textit{J. Bona} and \textit{Y. Lui} (cited in the references as a preprint). Using a methodology developed by the author [Trans. Am. Math. Soc. 353, 191--208 (2001; Zbl 0949.35120)], it is shown that, provided the initial conditions are close to an unstable solitary wave, the solution blows up in finite time if \(1\leq p<4/3\). This result improves on the prior result of \textit{J. Saut} [Acta Appl. Math. 39, 477--487 (1995; Zbl 0839.35121)] which established a corresponding result for \(p\geq 2\) (and for negative energy only).
reviewed by: A. Parker
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 37K45
Mathematics Subject Classification ID: 35Q51
Mathematics Subject Classification ID: 35B35
zbMATH DE Number: 1797539
zbMATH Keywords: solitary waves in three-dimensions
zbMATH Keywords: blow-up
MaRDI profile type: Publication
full work available at URL: jdeq.2001.4054
OpenAlex ID: W1980667375
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DOI: 10.1006/JDEQ.2001.4054
description: scientific article; zbMATH DE number 1797539