Algebraic-geometrical methods in the theory of integrable equations and their perturbations
type: Item
instance of: scholarly article
title: Algebraic-geometrical methods in the theory of integrable equations and their perturbations
zbMATH Open document ID: 0840.35095
DOI: 10.1007/BF00994629
author: Igor Krichever
published in: Acta Applicandae Mathematicae
publication date: 1996-07-04T00:00:00Z
review text: This is a review of basic techniques for applications of the algebraic-geometric methods to construction of vast classes of exact solutions to some two-dimensional nonlinear evolutional equations, first of all the Kadomtsev-Petviashvili (KP) equations, with periodic boundary conditions (b.c.). The methods are based on consideration of analytical properties of the Bloch eigenfunctions of the two-dimensional linear differential operators with the periodic b.c., that furnish the Lax-pair representation of the corresponding (formally) integrable equations, as functions of a complex spectral parameter (``energy''). The algebraic geometric construction is purely local. It maps a set of the so-called algebraic-geometric data into an exact solution to the nonlinear PDE. Eventually, the exact solutions prove to be finite-gap ones, which can be represented in terms of the corresponding Riemann's theta-function. All the exact solutions are quasi-periodic in time. For the KP-II equation, it can be proved that the set of these exact solutions is dense in the space of all the solutions with the periodic b.c., but this is not true for the KP-I equation, which is actually nonintegrable despite seeming formally integrable. It remains, however, unknown in what exact sense the KP-I equation is really nonintegrable. In the last chapter of this paper, a general scheme of the perturbation theory based on the algebraic-geometric techniques is outlined. It allows, e.g., to consider, in the spirit of the Witham's theory, slow modulations of the exact (unperturbed) quasiperiodic.
reviewed by: B. A. Malomed
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 35C05
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 37K10
zbMATH DE Number: 790793
zbMATH Keywords: Kadomtsev-Petviashvili equations
zbMATH Keywords: finite-gap solutions
zbMATH Keywords: exact solutions
zbMATH Keywords: Riemann's theta-function
zbMATH Keywords: quasiperiodic solutions
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00994629
OpenAlex ID: W4241521229
description: scientific article; zbMATH DE number 790793