Theoretical and Mathematical Physics, 192(1): 982–999 (2017)
QUASIDETERMINANT SOLUTIONS OF THE EXTENDED
NONCOMMUTATIVE KADOMTSEV–PETVIASHVILI HIERARCHY
and Chunxia Li
We construct a nonauto Darboux transformation for the extended noncommutative Kadomtsev–Petviash-
vili (ncKP) hierarchy and consequently derive its quasi-Wronskian solution. We also obtain the quasi-
Wronskian solution of the ncKP equation with self-consistent sources (ncKPESCS) as a by-product. Fi-
nally, we use the direct veriﬁcation method to prove the quasi-Wronskian solution of the ncKPESCS.
Keywords: extended noncommutative KP hierarchy, Darboux transformation, quasi-Wronskian solution,
It is known that the noncommutative (NC) gauge theory has some important applications in D-brane
theory. It was previously shown that a NC extension corresponds to the presence of background magnetic
ﬁelds and that NC solitons in some situations are themselves just lower-dimensional D-branes –.
In connection with this, extensions of classical integrable systems to their NC counterparts have attracted
much attention from many researchers . Some integrable-like properties of NC systems have been revealed
such as multisolitons –, Hamiltonian structures , B¨acklund transformations , inﬁnite conservation
laws , recursion operators , reductions , etc. Generally speaking, solutions of NC integrable systems
can be expressed in terms of quasideterminants . The authors of  used quasideterminants to integrate
the non-Abelian Toda ﬁeld equation for root systems of types A, B,andC. Maximally non-Abelian Toda
systems based on classical semisimple Lie groups were considered in detail in .
In 2007, Gilson and Nimmo used the Darboux transformation (DT) and binary DT to investigate
the NC Kadomtsev–Petviashvili (ncKP) and the NC modiﬁed Kadomtsev–Petviashvili (ncmKP) equa-
tions , . Two kinds of quasideterminant solutions, i.e., quasi-Wronskian and quasi-Grammian so-
lutions were presented. These quasideterminant solutions were then veriﬁed directly using the derivative
Department of Mathematics, Jimei University, Xiamen, China, e-mail: email@example.com,
School of Mathematical and Statistical Sciences, The University of Texas Rio Grande Valley, Edinburg, Texas,
School of Mathematical Sciences, Capital Normal University, Beijing, China, e-mail: trisha firstname.lastname@example.org.
This research is supported by the National Natural Science Foundation of China (Grant Nos. 11201178 and
11271266), the Fujian National Science Foundation (Grant No. 2012J01013), the Beijing National Science Foundation
(Grant No. 1162003), the Fujian Higher College Special Project of Scientiﬁc Research (Grant No. JK2012025), and
the Fujian Provincial Visiting Scholar Program.
Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i
Matematicheskaya Fizika, Vol. 192, No. 1, pp. 51–69, July, 2017. Original article submitted May 24, 2016; revised
July 30, 2016.
2017 Pleiades Publishing, Ltd.