A Generalized Dressing Approach for Solving the Extended KP and the Extended mKP Hierarchy

TL;DR

This paper introduces a generalized dressing method to solve extended KP and mKP hierarchies, providing explicit solutions and transformations between them, advancing integrable systems theory.

Contribution

It develops a unified dressing approach for extended KP and mKP hierarchies, including explicit Wronskian solutions and gauge transformations.

Findings

Constructed explicit Wronskian solutions for extended KP and mKP hierarchies.

Formulated the zero curvature and Lax representations for the extended hierarchies.

Established gauge transformations linking solutions of extended KP and mKP hierarchies.

Abstract

A combination of dressing method and variation of constants as well as a formula for constructing the eigenfunction is used to solve the extended KP hierarchy, which is a hierarchy with one more series of time-flow and based on the symmetry constaint of KP hierarchy. Similarly, extended mKP hierarchy is formulated and its zero curvature form, Lax representation and reductions are presented. Via gauge transformation, it is easy to transform dressing solutions of extended KP hierarchy to the solutions of extended mKP hierarchy. Wronskian solutions of extended KP and extended mKP hierarchies are constructed explicitly.