Recent results on the generalized Kadomtsev-Petviashvili equations
type: Item
instance of: scholarly article
title: Recent results on the generalized Kadomtsev-Petviashvili equations
zbMATH Open document ID: 0839.35121
DOI: 10.1007/BF00994650
published in: Acta Applicandae Mathematicae
publication date: 1996-06-25T00:00:00Z
review text: This paper presents a concise review of rigorous results obtained for the generalized Kadomtsev-Petviashvili (KP) equations in two dimensions, \[ u_t+ f(u) u_x+ u_{xxx}+ \varepsilon v_y= 0,\quad v_x= u_y,\tag{1} \] (where \(\varepsilon= \pm 1\)) and for similar equations in three dimensions. The power nonlinearities, \(f(u)= u^p\), are considered. It is known that the two-dimensional KP equations with \(p= 1\) are integrable by means of the inverse scattering transform. In this context, the KP equations with \(\varepsilon= -1\) and \(+1\) are called, respectively, the KP-I and KP-II equations. Nevertheless, even in this special case traditional methods of the PDE theory prove to be more powerful than the inverse scattering for proving the basic theorems about existence and uniqueness of solutions to the Cauchy problem for the KP equations. In particular, the global well-posedness of the Cauchy problem is proven for the integrable KP-II case, while for the KP-I case it is proven only for small initial data. Another important set of results is known for blowup (collapse) of the solutions, i.e., singularity formation in a finite time. While for the KP-II type equations this problem remains open, for the KP-I type using the so-called virial theorem (in two dimensions) has demonstrated that the blowup does take place if \(p\geq 4/3\), and does not happen in the opposite case. The final group of results pertains to existence and stability of solitons, i.e., solutions localized in all the spatial directions (note that the multidimensional KP solitons are localized not exponentially but only algebraically). It proves that the soliton solutions to the generalized KP equations can be completely classified. In particular, a soliton exists as a solution to the three-dimensional KP-I with \(p= 1\), but it is unstable. Generally, instability of the solitons is closely linked to the possibility of the blowup in the system.
reviewed by: B. A. Malomed
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 35A07
Mathematics Subject Classification ID: 35-02
zbMATH DE Number: 790814
zbMATH Keywords: review
zbMATH Keywords: inverse scattering transform
zbMATH Keywords: existence
zbMATH Keywords: uniqueness
zbMATH Keywords: global well-posedness of the Cauchy problem
zbMATH Keywords: blowup
zbMATH Keywords: stability of solitons
author: Jean-Claude Saut
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00994650
OpenAlex ID: W4255313102
description: scientific article; zbMATH DE number 790814