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

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.

💡 Research Summary

The paper presents a unified method for solving the extended Kadomtsev‑Petviashvili (KP) hierarchy and its modified counterpart (mKP) by extending the classical dressing technique with a variation‑of‑constants approach. The authors first introduce an additional time flow τₖ to the standard KP hierarchy, defining the extended KP hierarchy (exKPH) through the Lax equation
∂ₜₙL =