Abstract:
We tackle the problem of interpreting the Darboux transformation for the KP hierarchy and its relations with the modified KP hierarchy from a geometric point of view. This is achieved by introducing the concept of a Darboux covering. We construct a Darboux covering of the KP equations and obtain a new hierarchy of equations, which we call the Darboux-KP hierarchy (DKP). We employ the DKP equations to discuss the relationships among the KP equations, the modified KP equations, and the discrete KP equations. Our approach also handles the various reductions of the KP hierarchy.
We show that the KP hierarchy is a projection of the DKP, the mKP hierarchy is a DKP restriction to a suitable invariant submanifold, and that the discrete KP equations are obtained as iterations of the DKP ones.
Similar content being viewed by others
Author information
Authors and Affiliations
Additional information
Received: 23 July 1996 / Accepted: 6 January 1997
Rights and permissions
About this article
Cite this article
Magri, F., Pedroni, M. & Zubelli, J. On the Geometry of Darboux Transformations for the KP Hierarchy and its Connection with the Discrete KP Hierarchy . Comm Math Phys 188, 305–325 (1997). https://doi.org/10.1007/s002200050166
Issue Date:
DOI: https://doi.org/10.1007/s002200050166