The fermionic approach to Darboux transformations

Abstract

Starting from the free fermion description of the one-component KP hierarchy, we establish a connection between this approach and the theory of Darboux and binary Darboux transformations. Certain difference identities - allowing for the treatment of both continuous as well as discrete evolution equations - turn out to be crucial: first to show that any solution of the associated (adjoint) linear problems can always be expressed as a superposition of KP (adjoint) wavefunctions and then to interpret Darboux (and binary Darboux) transformations as Bäcklund transformations in the fermion language.