Construction of higher-dimensional nonlinear integrable systems and of their solutions
type: Item
instance of: scholarly article
title: Construction of higher-dimensional nonlinear integrable systems and of their solutions
zbMATH Open document ID: 0597.35115
DOI: 10.1007/BF01078388
author: S. V. Manakov
author: V. E. Zakharov
publication date: 1985-01-01T00:00:00Z
review text: On the basis of the nonlocal Riemann problem and nonlocal \({\bar \partial}\)-problem a systematic exposition is given of the procedure of constructing higher dimensional integrable nonlinear equations. The nonlocal Riemann problem is to find an \(N\times N\) matrix function \(\chi\) (\(\lambda)\) analytic everywhere off the contour \(\Gamma\), whose boundary values \(\chi_ 1\) and \(\chi_ 2\) on \(\Gamma\) are connected by the integral relation \[ \chi_ 2(\lambda)=\chi_ 1(\lambda)+\int_{\Gamma}\chi_ 1(\lambda ')T(\lambda ',\lambda)d\lambda '. \] It is supposed that T(\(\lambda\),\(\lambda\) ') depends on \(n\geq 2\) supplementary variables \(x_ i\) in such a way that \[ \partial T(\lambda ',\lambda)/\partial x_ i=I_ i(\lambda ')T(\lambda ',\lambda)-T(\lambda ',\lambda)I_ i(\lambda), \] where \(I_ i(\lambda)\) are pairwise-commuting matrix-valued rational functions of the parameter \(\lambda\). It is shown that in this case the solution \(\chi\) of the Riemann problem satisfies the system of linear differential equations generating the corresponding nonlinear equations. The suggested procedure of constructing nonlinear equations and their solutions remains the same if one uses the more generalized nonlocal \({\bar \partial}\)- problem \[ 2i \partial \chi (\lambda,{\bar \lambda})/\partial {\bar \lambda}=\int \chi (\lambda ',{\bar \lambda}') R(\lambda ',{\bar \lambda}',\lambda,\quad {\bar \lambda}) d\lambda ' d{\bar \lambda}' \] where \(\partial R/\partial x_ i=I_ i(\lambda ') R(\lambda ',{\bar \lambda}',\lambda,{\bar \lambda})-R(\lambda ',{\bar \lambda}',\lambda,\lambda \bar {\;}) I_ i(\lambda).\) The effectiveness of the developed methods are illustrated on the Kadomtsev-Petviashvili equation and the Korteweg-de Vries equation.
Mathematics Subject Classification ID: 35Q99
Mathematics Subject Classification ID: 35N15
zbMATH DE Number: 3962351
zbMATH Keywords: nonlinear integrable systems
zbMATH Keywords: nonlocal Riemann problem
zbMATH Keywords: nonlocal \({\bar \partial }\)-problem
zbMATH Keywords: Kadomtsev-Petviashvili equation
zbMATH Keywords: Korteweg-de Vries equation
MaRDI profile type: Publication
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published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 3962351