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Holomorphic vector bundles over Riemann surfaces and the Kadomtsev-Petviashvili equation. I. (Russian) Zbl 0393.35061
The paper starts with the Kadomtsev-Petviashvili equation, written in the form \[ 0=\frac34 \frac{\partial^2U}{\partial y^2}+\frac{\partial}{\partial x}\left[\frac{\partial U}{\partial t}+\frac14 \left(6U\frac{\partial U}{\partial x}+\frac{\partial^3U}{\partial x^3}\right)\right], \] for which the second author [Funct. Anal. Appl. 8, 236–246 (1974); translation from Funkts. Anal. Prilozh. 8, No. 3, 54–66 (1974; Zbl 0299.35017)] and V. E. Zakharov have given solutions generalizing those of the Korteweg-de Vries equation. The aim of the present paper is to find solutions depending on arbitrary functions. By aid of Baker-Akhiezer’s functions corresponding to a set of matrices \(\psi_0\), to an algebraic curve \(\Gamma\), and a point \(P_0\in\Gamma\), one can construct a solution of (1). For particular cases, each solution of the Korteweg-de Vries equation or the Boussinesq equations, respectively generates solutions of the Kadomtsev-Petviashvili equation.
Reviewer: A. Haimovici

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
32L05 Holomorphic bundles and generalizations
37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions
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