Two choices of the gauge transformation for the AKNS hierarchy through the constrained KP hierarchy

Abstract

On the basis of the equivalence between the AKNS hierarchy and the cKP hierarchy with the constraint k=1, we point out that there exist two choices to keep the form of the Lax operator when we perform the gauge transformation for the AKNS hierarchy, which results in two classes of functions to trigger the gauge transformation. For the second choice, two theorems for two types of gauge transformation are established. Several new and more general forms of tau-functions for the AKNS hierarchy are obtained by means of gauge transformations of both types. The union of the two choices leads to new forms of τ-functions. We generate the AKNS hierarchy from the “free” Lax operator L(0)=∂ via a chain of gauge transformations.