Bilinear Identities and Hirota's Bilinear Forms for an Extended Kadomtsev-Petviashvili Hierarchy

TL;DR

This paper develops bilinear identities and Hirota's bilinear forms for an extended KP hierarchy, incorporating auxiliary parameters and symmetries, leading to simplified equations for KP equations with self-consistent sources.

Contribution

It introduces a new approach to derive bilinear identities for an extended KP hierarchy using an auxiliary parameter, simplifying the Hirota equations for KP with sources.

Findings

Derived bilinear identities for the extended KP hierarchy.

Established Hirota's bilinear equations for KP with self-consistent sources.

Demonstrated the simplified form of these equations compared to previous results.

Abstract

In this paper, we construct the bilinear identities for the wave functions of an extended Kadomtsev-Petviashvili (KP) hierarchy, which is the KP hierarchy with particular extended flows (2008, Phys. Lett. A, 372: 3819). By introducing an auxiliary parameter (denoted by $z$), whose flow corresponds to the so-called squared eigenfunction symmetry of KP hierarchy, we find the tau-function for this extended KP hierarchy. It is shown that the bilinear identities will generate all the Hirota's bilinear equations for the zero-curvature forms of the extended KP hierarchy, which includes two types of KP equation with self-consistent sources (KPSCS). It seems that the Hirota's bilinear equations obtained in this paper for KPSCS are in a simpler form by comparing with the results by Hu and Wang (2007, Inverse Problems, 23: 1433).