Physics Letters A 372 (2008) 5523–5528
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Physics Letters A
www.elsevier.com/locate/pla
Darboux transformation for the non-isospectral AKNS hierarchy
and its asymptotic property
Lingjun Zhou
Department of Mathematics, Tongji University, Shanghai 200092, PR China
article info abstract
Article history:
Received 17 May 2007
Received in revised form 19 April 2008
Accepted 4 June 2008
Available online 2 July 2008
Communicated by A.P. Fordy
PACS:
02.30.Ik
02.30.Jr
MSC:
35Q53
35Q55
Keywords:
Non-isospectral AKNS hierarchy
Darboux transformation
Integral constant
In this Letter, the Darboux transformation for the non-isospectral AKNS hierarchy is constructed. We
show that the Darboux transformation for the non-isospectral AKNS hierarchy is not an auto-Bäcklund
transformation, because the integral constants of the hierarchy will be changed after the transformation.
The transform rule of the integral constants will be also derived. By this means, the soliton solutions of
the nonlinear equations derived by the non-isospectral AKNS hierarchy can be found.
©
2008 Elsevier B.V. All rights reserved.
1. Introduction
The Darboux matrix formalism was introduced in connection
with the dressing method as originated by Zakharov and Shabat
in [1] and described in the text [2]. Important work on the Dar-
boux matrix method was conducted, on the one hand, by Matveev
and Salle [3] and on the other hand, by Neugebauer and Meinel [4].
Links between Bäcklund transformation and the dressing method,
as well as between the latter and the classical Darboux transfor-
mation, have been elucidated by Levi et al. [5,6]. The AKNS hier-
archy is one of the most important integrable systems, which was
first introduced by Ablowitz, Kaup, Newell, and Segur [7].Many
important nonlinear differential equation are equivalent to the in-
tegrability condition of AKNS hierarchy [8–10]. A unified approach
to construct Darboux transformations with matrix form for AKNS
hierarchy was founded by Gu [11–16], in which the geometric ap-
plications of the Darboux transformation are also elucidated.
Non-isospectral problem is the problem in which the spectral
parameter depend on the time or space variables. The Ernst Equa-
tions [17,18], which are the Einstein’s equations for axially sym-
E-mail address: zhoulj@mail.tongji.edu.cn.
metric gravitational fields, were represented as the zero-curvature
equations of non-isospectral problems by Maison [19], while Har-
rison derived a Bäcklund transformation for the Ernst equation of
general relativity by using the Wahlquist–Estabrook procedure [20].
In 1979, Kramer and Neugebauer also independently established
the Bäcklund transformation for Ernst equation [21]. An excellent
account of the Darboux matrix method with particular attention to
non-isospectral problems has been given by Cie
´
sli
´
nski [22].
In this letter, we will construct the non-isospectral AKNS hier-
archy and its Darboux transformation. The method for the standard
AKNS hierarchy in the literatures [11–16] can be generalized to the
non-isospectral case [23]. However, the Darboux transformation
for the non-isospectral AKNS hierarchy has an essential difference
from the standard case, that its integral constants may not be con-
served by the transform. So one may not get the nontrivial soliton
solution of the relevant nonlinear equation by acting the Darboux
transformation on the seed solution of the same equation [24,25].
In this article, we will prove that the relation of the integral con-
stants between the relevant non-isospectral AKNS hierarchies can
be calculated through the asymptotic property of the elementary
solution. Then the soliton solution of a certain differential equa-
tion can be found by acting the Darboux transformation on the
seed solution of another equation.
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©
2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.physleta.2008.06.072