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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

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.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

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

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

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

(n)

= I∂

+ A

(n)

, L

(n)

= I∂

+ 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)

(n)

= L

(n)

G ⇒ ∂

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



(n)

, D

(m)



= 0, then the trans-

formation (2.2) leads to the generalized Zakharov-Shabat zero curvature relations, which deﬁne

the integrable hierarchy

∂

(m)

− ∂

(n)



(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

be the solution of the linear problem:

(n)

= D

(n)

, (2.4)

and G is the generic Darboux transform:

= 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

= 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) =



F(x,y)f (y) dy,

(f )(x) =



(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

+ 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:

(x, z) + F(x,z)+

∞



dy K

(x, y)F (y, z) = 0,z>x,

−

(x, z) = F(x,z)+

∞



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

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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