Algebraic-geometrical methods in the theory of integrable equations and their perturbations
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.
scientific article; zbMATH DE number 790793
scientific article