Construction of higher-dimensional nonlinear integrable systems and of their solutions
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.
scientific article; zbMATH DE number 3962351
scientific article