42 research outputs found
Restricted Flows and the Soliton Equation with Self-Consistent Sources
The KdV equation is used as an example to illustrate the relation between the
restricted flows and the soliton equation with self-consistent sources.
Inspired by the results on the Backlund transformation for the restricted flows
(by V.B. Kuznetsov et al.), we constructed two types of Darboux transformations
for the KdV equation with self-consistent sources (KdVES). These Darboux
transformations are used to get some explicit solutions of the KdVES, which
include soliton, rational, positon, and negaton solutions.Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on
Integrable Systems and Related Topics, published in SIGMA (Symmetry,
Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA
Bilinear Identities and Hirota's Bilinear Forms for an Extended Kadomtsev-Petviashvili Hierarchy
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 ), 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).Comment: 23 pages, submitted to JNM
The q-deformed mKP hierarchy with self-consistent sources, Wronskian solutions and solitons
Based on the eigenfunction symmetry constraint of the q-deformed modified KP hierarchy, a q-deformed mKP hierarchy with self-consistent sources (q-mKPHSCSs) is constructed. The q-mKPHSCSs contain two types of q-deformed mKP equation with self-consistent sources. By the combination of the dressing method and the method of variation of constants, a generalized dressing approach is proposed to solve the q-deformed KP hierarchy with self-consistent sources (q-KPHSCSs). Using the gauge transformation between the q-KPHSCSs and the q-mKPHSCSs, the q-deformed Wronskian solutions for the q-KPHSCSs and the q-mKPHSCSs are obtained. The one-soliton solutions for the q-deformed KP (mKP) equation with a source are given explicitly
The relation between the Toda hierarchy and the KdV hierarchy
Under three relations connecting the field variables of Toda flows and that
of KdV flows, we present three new sequences of combination of the equations in
the Toda hierarchy which have the KdV hierarchy as a continuous limit. The
relation between the Poisson structures of the KdV hierarchy and the Toda
hierarchy in continuous limit is also studied.Comment: 11 pages, Tex, no figures, to be published in Physics Letters
A new extended KP hierarchy
A method is proposed to construct a new extended KP hierarchy, which includes
two types of KP equation with self-consistent sources and admits reductions to
k-constrained KP hierarchy and to Gelfand-Dickey hierarchy with sources. It
provides a general way to construct soliton equations with sources and their
Lax representations.Comment: Published in Phys. Lett. A, 13 page