Darboux theorems and Wronskian formulas for integrable systems: I. Constrained KP flows

Abstract

Generalizations of the classical Darboux theorem are established for pseudo-differential scattering operators of the form L = ∑ i=0 N u i∂ i + Σ i=1 m Φ i∂ −1 Ψ i † i . Iteration of the Darboux transformations leads to a gauge transformed operator with coefficients given by Wronskian formulas involving a set of eigenfunctions of L. Nonlinear integrable partial differential equations are associated with the scattering operator L which arise as a symmetry reduction of the multicomponent KP hierarchy. With a suitable linear time evolution for the eigenfunctions the Darboux transformation is used to obtain solutions of the integrable equations in terms of Wronskian determinants.