Commuting Differential Operators of Rank 3, and Nonlinear Differential Equations

Complete solutions of the commutation equations of ordinary differential operators are obtained, to which there corresponds a three-dimensional vector bundle of common eigenfunctions over an elliptic curve. The deformation of the commuting pair by the Kadomtsev–Petviashvili equation is studied. The finite-zone solutions of the Kadomtsev–Petviashvili equation of rank 3 and genus 1 are explicitly expressed in terms of functional parameters satisfying a Boussinesq-type system of two evolution equations. Bibliography: 40 titles.