Instability of solitons for the critical generalized Korteweg-de Vries equation
type: Item
instance of: scholarly article
title: Instability of solitons for the critical generalized Korteweg-de Vries equation
zbMATH Open document ID: 0985.35071
DOI: 10.1007/PL00001673
author: Yvan Martel
author: Frank Merle
published in: Geometric and Functional Analysis. GAFA
publication date: 2002-05-22T00:00:00Z
review text: The paper considers stability of soliton solutions to the generalized KdV equation of the form \[ u_t + u_{xxx} +(u^5)_x = 0. \] This model is called a critical one, as, with power 5 of the nonlinear term in the equation, it exactly falls on the boundary between equations of the KdV type with weaker nonlinearities, which give rise to stable solitons, and equations with stronger nonlinearities, in which strong collapse (formation of a singularity in finite time) takes place, and solitons are unstable, so that a small perturbation provokes the onset of collapse of a soliton. Numerical simulations, reported in earlier works, demonstrated instability of solitons in the critical equation. In the critical case, the equation gives rise to weak collapse (i.e., the onset of the collapse depends on the initial configuration). In accordance with this, numerical simulations show that the instability of solitons in the critical equation leads to formation of a singularity. In this paper, the instability of solitons in the critical KdV equation is proved rigorously by means of variational estimates of the virial type.
reviewed by: B. A. Malomed
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 37K45
zbMATH DE Number: 1618834
zbMATH Keywords: weak collapse
zbMATH Keywords: virial estimate
MaRDI profile type: Publication
full work available at URL: pl00001673
OpenAlex ID: W2067674243
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description: scientific article; zbMATH DE number 1618834