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我们考虑广义矩阵非线性薛定ding(NLS)层次结构。 通过采用通用的Darboux拟合方案,我们通过矩阵Gelfand-Levitan-Marchenko方程的解来导出可积PDE层次的解,并且我们还确定了递归关系,从而产生了整个矩阵NLS型层次的Lax对。 这些结果是通过将矩阵积分或一般的n阶矩阵微分算子视为Darboux-dressing变换而获得的。 在此框架中,还讨论了与Airy和Burgers方程的特殊链接。 还检查了Darboux变换的矩阵形式,从而得出Riccati方程的非交换形式。 求解了非可交换Riccati方程,因此得出了合适的守恒量。 在这种情况下,我们还讨论了NLS矩阵模型的无穷维情况,因为它为常规NLS模型的量子形式提供了合适的候选对象。 类似地,推导了用于一般修整变换的非交换Riccati方程,它自然等效于从辅助线性问题的解中得出的方程。
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Available online at www.sciencedirect.com
ScienceDirect
Nuclear Physics B 941 (2019) 376–400
www.elsevier.com/locate/nuclphysb
Non-commutative NLS-type hierarchies:
Dressing & solutions
Anastasia Doikou
∗
, Iain Findlay, Spyridoula Sklaveniti
School of Mathematical and Computer Sciences, Department of Mathematics, Heriot-Watt University,
Edinburgh EH14 4AS, United Kingdom
Received 28
October 2018; received in revised form 13 February 2019; accepted 17 February 2019
Available
online 26 February 2019
Editor: Hubert
Saleur
Abstract
We
consider the generalized matrix non-linear Schrödinger (NLS) hierarchy. By employing the univer-
sal
Darboux-dressing scheme we derive solutions for the hierarchy of integrable PDEs via solutions of
the matrix Gelfand-Levitan-Marchenko equation, and we also identify recursion relations that yield the
Lax pairs for the whole matrix NLS-type hierarchy. These results are obtained considering either matrix-
integral
or general n-th order matrix-differential operators as Darboux-dressing transformations. In this
framework special links with the Airy and Burgers equations are also discussed. The matrix version of the
Darboux transform is also examined leading to the non-commutative version of the Riccati equation. The
non-commutative Riccati equation is solved and hence suitable conserved quantities are derived. In this
context we also discuss the infinite dimensional case of the NLS matrix model as it provides a suitable
candidate for a quantum version of the usual NLS model. Similarly, the non-commutative Riccati equation
for the general dressing transform is derived and it is naturally equivalent to the one emerging from the
solution of the auxiliary linear problem.
© 2019 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
*
Corresponding author.
E-mail
addresses: a.doikou@hw.ac.uk (A. Doikou), iaf1@hw.ac.uk (I. Findlay), ss153@hw.ac.uk (S. Sklaveniti).
https://doi.org/10.1016/j.nuclphysb.2019.02.019
0550-3213/© 2019
The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
A. Doikou et al. / Nuclear Physics B 941 (2019) 376–400 377
1. Introduction
Non-linear Schrödinger-type models (AKNS scheme) are among the most well studied pro-
totypical integrable hierarchies (see for instance [1–12] and references therein). Both continuum
and discrete versions have been widely studied from the point of view of the inverse scatter-
ing method or the Darboux and Zakharov-Shabat (ZS) dressing methods [13–17]. W
ithin the ZS
scheme [14] solutions of integrable non-linear PDEs can be obtained by means of solutions of the
associated linear problem, and also the hierarchy of Lax operators can be explicitly constructed.
Similarly, various studies from the Hamiltonian point of view [15]in the case of periodic as well
as generic inte
grable boundary conditions (see e.g. [18,11]) exist. In the algebraic scheme the
hierarchy of Lax pairs can also be constructed via the universal formula based on the existence
of a classical r-matrix [19], extended also to the case of integrable boundary conditions [20], as
well as at the quantum le
vel [21].
We consider in this study a generalized matrix ve
rsion of an NLS-type model (matrix AKNS
version). This investigation can be divided into two main parts:
1. We in
vestigate solutions of the generalized matrix NLS hierarchy based on the generic no-
tion of Darboux-dressing transformation. We explicitly derive recursion relations that yield
the Lax pairs for the whole hierarchy, and we also obtain generalized solutions by employ-
ing both discrete and continuous solutions of the associated linear problem and solving the
matrix (non-commutati
ve) Gelfand-Levitan-Marchenko (GLM) equation. Solutions of the
generalized mKdV model in particular, a member of the hierarchy under study, are derived
and expressed in terms of Airy functions, generalizing the results in [22].
2. We study the non-commutati
ve Riccati flows for the matrix NLS hierarchy, aiming primarily
at deriving the corresponding hierarchy of conserved quantities. In particular, we derive and
solve the non-commutative Riccati equation, and hence we obtain the auxiliary function and
relevant classical and non-commutative conserved quantities. In this context the matrix NLS
model can be seen as a non-commutati
ve version of the familiar NLS model, and be further
upgraded to the quantum version of the usual NLS model provided that suitable commutation
relations are imposed among the non-commutative fields. It is worth noting that relevant
studies on the formulation of the quantum GLM equation for the NLS model were discussed
in [23]. The dressing transform is also performed in the case of Lax pairs e
xpressed in a
matrix form, and the non-commutative Riccati equation for the generic Darboux-dressing is
identified.
Let us summarize belo
w what is achieved in this investigation. In the first part of this study
(sections 2 and 3) we implement the generalized Darboux-dressing method in order to derive
solutions as well as recursion relations that produce the Lax pairs for the whole hierarchy. The
Darboux transform in our frame is chosen to be either an inte
gral or a differential operator. When
choosing the Darboux transform to be an integral operator we basically apply the Zakharov-
Shabat (ZS) dressing method as described in [14]. One of the main advantages of the ZS dressing
method is that no analyticity conditions are ap
riorirequired for the solutions of the correspond-
ing scattering problem, as is the case when applying the inverse scattering transform, thus this
method provides essentially a linearization of non-linear ODEs and PDEs. Here, based on the
universal Darboux-dressing scheme we produce solitonic solutions as well as formal expressions
for generic solutions via the Fredholm theory (see also [22]), and also deri
ve sets of recursion
relations that yield the Lax pairs for the integrable hierarchy at hand. Moreover, by exploiting
378 A. Doikou et al. / Nuclear Physics B 941 (2019) 376–400
the cubic dispersion relations for the third member of the hierarchy, i.e. the generalized matrix
mKdV model, we are able to express the formal solutions in terms of Airy functions, and show
that solutions of the corresponding linear problem satisfy the Airy equation, which is known to
be a linearization of P
ainlevé II. Explicit expressions of the dressed Lax pairs for the first few
members of the hierarchy are also reported. The viscous and inviscid matrix Burgers equations
are also derived as special cases in our setting.
In section 4 we in
vestigate non-commutative Riccati flows associated to the matrix NLS hier-
archy; these are also directly related to the notion of Grassmannian flows studied in the context
of non-local, non-linear PDEs in [24,25]. We solve the non-commutative Riccati equation and
subsequently derive suitable conserved quantities based on the solution of both time and space
Riccati flo
ws. A brief discussion on the relevance of the quantum version of NLS with our find-
ings is also presented. The Poisson structure for the components of the matrix fields, is identified
via the comparison of the equations of motion from the zero curvature condition and via the
Hamiltonian formulation. The dressing transform is also performed in the case of Lax pairs ex
-
pressed in a matrix form, and the equivalence of this description with the case of differential
operators as dressing transforms is displayed. Moreover, we consider the generic dressing trans-
form expressed as a formal series e
xpansion as well as an integral operator and identify the
associated non-commutative Riccati equation, which is naturally equivalent to the one derived
for the solution of the space part of the linear auxiliary problem. The findings of this section
are closely related to the results presented in [21]
regarding the ideas on the quantum auxiliary
linear problem and the quantum Darboux-Bäcklund transforms. Note that in [21]a certain ver-
sion of the quantum discrete NLS model was studied, i.e. the quantum Ablowitz-Ladik model,
so comparisons with the present findings can be made.
2. Dressing transformations as integral operators
The main aims of this section are: 1) the derivation of solutions of the integrable non-linear
PDEs for the hierarchy under study and 2) the construction of the Lax pairs of all the members
of the matrix NLS (AKNS) hierarchy. The key idea is that solutions of the non-linear integrable
PDEs are obtained via the matrix Gelf
and-Levitan-Marchenko equation, which will be derived
below, by employing solutions of the associated linear problem.
We re
view now the main ideas of the generic Darboux-dressing scheme [14]. Let
D
(n)
= I∂
t
n
+ A
(n)
, L
(n)
= I∂
t
n
+ A
(n)
, (2.1)
where I in the N × N unit matrix and A
(n)
, A
(n)
are some “bare” (free of fields) and “dressed”
operators respectively. In general, they can be N × N matrices, matrix-differential or matrix-
integral operators, and are related via the generic Darboux-dressing transformation (which is a
typical similarity transformation)
GD
(n)
= L
(n)
G ⇒ ∂
t
n
G = GA
(n)
− A
(n)
G. (2.2)
It is worth noting that this fundamental idea was generalized in [24,25] and led to the construction
of non-local non-linear PDEs as well as to their solution via solutions of the associated linear
problem.
We consider bare dif
ferential operators D
(n)
that commute
D
(n)
, D
(m)
= 0, then the trans-
formation (2.2) leads to the generalized Zakharov-Shabat zero curvature relations, which define
the integrable hierarchy
∂
t
n
A
(m)
− ∂
t
m
A
(n)
+
A
(n)
, A
(m)
= 0. (2.3)
A. Doikou et al. / Nuclear Physics B 941 (2019) 376–400 379
In our setting here we choose to consider for now A
(n)
, A
(n)
as matrix-differential operators,
whereas the dressing transform G is chosen to be either a matrix-integral operator or a matrix-
differential one.
We first consider the case where the Darboux transform G is an inte
gral operator, and we
derive the matrix GLM equation from the fundamental factorization condition (or the general
Darboux transform). Let Q
0
be the solution of the linear problem:
Q
0
D
(n)
= D
(n)
Q
0
, (2.4)
and G is the generic Darboux transform:
GQ
0
= Q. (2.5)
The two equations above lead to Q D
(n)
= L
(n)
Q, which together with (2.2) naturally suggest
the equivalence of the two generic objects G, Q. To ensure invertibility of the related operators
we express: G = I + K
+
, Q
0
= I + F, Q = I + K
−
, where K
±
, F have integral representations
(f is the test function, an N -column vector in general):
F(f )(x) =
R
F(x,y)f (y) dy,
K
±
(f )(x) =
R
K
±
(x, y)f (y) dy, (2.6)
such that: K
+
(x, y) = 0, x>y, and K
−
(x, y) = 0, x<y.
The operators K
±
, F then satisfy via (2.5)the factorization condition
K
+
+ F + K
+
F = K
−
, (2.7)
which leads to the fact that the kernel K
+
(x, y) satisfies the Gelfand-Levitan-Marchenko equa-
tion and K
−
(x, y) obeys an analogous integral equation:
K
+
(x, z) + F(x,z)+
∞
x
dy K
+
(x, y)F (y, z) = 0,z>x,
K
−
(x, z) = F(x,z)+
∞
x
dy K
+
(x, y)F (y, z), z < x. (2.8)
F(x, y) is the solution of the linear problem, i.e. invariance of the differential operators D
(n)
under the action of the operator F is required (2.4). Dependence on the universal time t that
includes all the times t
n
is implied but omitted for now for simplicity. Note that the factorization
condition (2.7)is nothing but the analogue of the Darboux-dressing transformation acting on the
linear solution F and providing the transformed solution K
−
. We shall address this issue again
when dealing with the auxiliary matrix problem in the matrix language. The kernel K
+
, as will
be transparent in the following, is the quantity that produces solutions of the integrable PDEs
emerging from the zero curvature condition. One may think of the GLM equation as a necessary
intermediate step between the linear problem and the non-linear integrable PDE.
It is wo
rth pointing out that the integral representations (2.6) lead to a Volterra type inte-
gral equation (GLM). In this frame the condition (2.5)is equivalent to a Borel decomposition,
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