Discrete symmetries and nonlocal reductions

Contents lists available atScienceDirect

Physics

Letters

A

www.elsevier.com/locate/pla

Discrete

symmetries

and

nonlocal

reductions

Metin Gürses

a

,

Aslı Pekcan

b

,

∗

,

Kostyantyn Zheltukhin

c

a_{Department of Mathematics, Faculty of Science, Bilkent University, 06800 Ankara, Turkey} b_{Department of Mathematics, Faculty of Science, Hacettepe University, 06800 Ankara, Turkey} c_{Department of Mathematics, Faculty of Science, Middle East Technical University, 06800 Ankara, Turkey}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Article history: Received8July2019

Receivedinrevisedform14October2019 Accepted15October2019

Availableonline21October2019 CommunicatedbyF.Porcelli Keywords: Integrablesystems Scalesymmetries Discretesymmetries Nonlocalreductions

Weshowthatnonlocalreductionsofsystemsofintegrablenonlinearpartialdifferentialequationsarethe specialdiscretesymmetrytransformations.

©2019ElsevierB.V.Allrightsreserved.

1. Introduction

Nonlocal reductionsof systems of integrablenonlinear partial differentialequations which were invented first by Ablowitz and Musslimani[1–3],attractedmanyresearchersinthefield.Ablowitz andMusslimanihavefirstconstructednonlocalreductionfor non-linearSchrödinger(NLS) systemofequationsandobtained nonlo-cal nonlinear Schrödinger (nNLS) equation [1], [2]. Theyshowed that nNLS equation is integrable, i.e., it admits a Lax pair, and foundsolitonsolutionsbytheuseoftheinversescatteringmethod. Ablowitz and Musslimani have later extended their nonlocal re-ductions, corresponding to space reflection, time reflection, and space-timereflectionto modified Korteweg-deVries (mKdV) sys-tem,sine-Gordon(SG)system, Davey-Stewartson(DS)system, and soon.AfterAblowitzandMusslimani’sworksthereisahuge inter-estinobtainingnonlocalreductionsofsystemsofintegrable equa-tionsandfindinginterestingwavesolutionsofthesesystems. Spe-cific examples are nonlocal NLS equation [1–14], nonlocal mKdV equation [2–4], [13], [15–18], nonlocal SG equation [2–4], [19], nonlocal DS equation [3], [20–24], nonlocal Fordy-Kulish equa-tions [13],[25],nonlocal N-wavesystems[3],[26],nonlocal vec-torNLS equations[27–30],nonlocal

(

2

+

1

)

-dimensional negative AKNSsystems[31], andnonlocalcoupledHirota-Iwao mKdV sys-tems[32].See[33] forthediscussionofsuperpositionofnonlocal

*

Correspondingauthor.

E-mail addresses:gurses@fen.bilkent.edu.tr(M. Gürses),

aslipekcan@hacettepe.edu.tr(A. Pekcan),

zheltukh@metu.edu.tr

(K. Zheltukhin).

integrable equations,and [34] for thenonlocal reductions of the integrable equations of hydrodynamic type. The connection be-tween local and nonlocal reductionsis given in [35], [36]. In all these works the soliton solutions and their properties were in-vestigated by usingthe inverse scatteringmethod, by the Hirota bilinearmethod,andbytheDarbouxtransformations.

In the last decade we observe that even as the number of systems of integrable nonlinear differential equations possessing nonlocalreductionsisincreasing,theoriginofnonlocalreductions remainsmysterious.Recently,Valchev[37] generalizingthe reduc-tion group ofMikhailov [38], (see also [6], [26], [29] for the ap-plicationofMikhailov’sreductiongroup)transformingalsothe in-dependentvariables obtainedthenonlocal reductionsofAblowitz andMusslimani.Inthisworkweaddresstothisproblem.Weshow that those systemspossessing nonlocal reductionsadmit discrete symmetry transformations which leave the systems invariant. A specialcaseofdiscretesymmetry transformationsturnsouttobe thenonlocalreductionsofthesamesystems.Weshowthisfactfor NLS, mKdV,SG,DS,coupled NLS-derivativeNLS,loop soliton sys-tems,hydrodynamictypesystems,andFordy-Kulishequations,and deriveallpossiblenonlocalreductionsfromthediscretesymmetry transformationsofthesesystems.

2. Reductions

Letthedynamicalvariablesqi

₍

_t

_,

_x

₎

_andri

₍

_t

_,

_x

₎

_(i

₌

₁

_,

₂

_,

_{· · · ,}

_N), in

(

1

+

1

)

-dimensions, satisfy the following system of integrable evolutionequations

https://doi.org/10.1016/j.physleta.2019.126065 0375-9601/©2019ElsevierB.V.Allrightsreserved.

q_ti

=

Fi

(

qj

,

rj

,

qxj

,

rxj

,

qxxj

,

rxxj

,

· · · ),

i

,

j

=

1

,

2

,

· · · ,

N

,

(1)

r_ti

=

Gi

(

qj

,

rj

,

qxj

,

rxj

,

qxxj

,

rxxj

,

· · · ),

i

,

j

=

1

,

2

,

· · · ,

N

,

(2) where Fi _{and G}i

₍

_i

₌

₁

_,

₂

_,

_{· · · ,}

_N

₎

_{are functionsofthedynamical} variablesqi

(

t

,

x

)

,ri

(

t

,

x

)

,andtheirpartialderivativeswithrespect to x.Theabove systemofequationsisintegrable, soit hasaLax pairand a recursionoperator

R

. Some oftheseequations admit localandnonlocalreductions.Letusassumethattheabovesystem ofequations(1) and(2) admitsthefollowingreductions.

(a) Local reductions:

Thelocalreductionsaregivenby

ri

(

t

,

x

)

=

κ

1qi

(

t

,

x

),

i

=

1

,

2

,

· · · ,

N

,

(3) and

ri

(

t

,

x

)

=

κ

2q

¯

i

(

t

,

x

),

i

=

1

,

2

,

· · · ,

N

,

(4) where

κ

1 and

κ

2 are realconstants.Throughout thispaperabar overaletterisdefinedas

1) foracomplexnumberq

=

α

+

i

β

,q

¯

=

α

−

i

β

,i2

_{= −}

_1, 2) forapseudo-complexnumberq

=

α

+

i

β

,q

¯

=

α

−

i

β

,i2

=

1. Ifareductionis consistentthesystemofequations(1) and (2) is reducedtoasystemforqi’s

q_ti

= ˜

Fi

(

qj

,

qxj

,

qxxj

,

· · · ),

i

,

j

=

1

,

2

,

· · · ,

N (5) forthereduction(3),and

q_ti

= ˜

Fi

(

qj

,

q

¯

j

,

qxj

,

q

¯

xj

,

qxxj

,

q

¯

xxj

,

· · · ),

i

,

j

=

1

,

2

,

· · · ,

N (6) forthereduction(4),where F

˜

=

F

|

_ri_=κ

1qi or F

˜

=

F

|

ri=κ2q¯i,

respec-tively.

(b) Nonlocal reductions:

Recently,AblowitzandMusslimaniintroducednewtypeof reduc-tions[1–3]

ri

(

t

,

x

)

=

τ

1qi

(

ε

1t

,

ε

2x

)

=

τ

1qiε

,

(7)

and

ri

(

t

,

x

)

=

τ

2q

¯

i

(

ε

1t

,

ε

2x

)

=

τ

2q

¯

iε

,

(8)

fori

=

1

,

2

,

· · · ,

N.Here

τ

1and

τ

2arerealconstantsand

ε

21

=

ε

22

=

1.

When

(

ε

1

,

ε

2

)

= (−

1

,

1

),

(

1

,

−

1

),

(

−

1

,

−

1

)

, the above con-straints reduce the system (1) and (2) to nonlocal space re-flection symmetric (S-symmetric), time reflection symmetric (T-symmetric), or space-time reflection symmetric (ST-symmetric) differentialequations.

Since the reductions are done consistently the reduced sys-temsofequationsarealsointegrable.Thismeansthatthereduced systems admit recursion operators and Lax pairs. We can obtain

N-solitonsolutionsofthereducedsystemsbytheinverse scatter-ingmethod[1–3],[10],[11],[14],[17],[19],[27],bytheDarboux transformation[9],[16],[18],[22],[23],andbytheHirotabilinear method[7],[13],[15],[21],[31–33].

3. Discrete symmetries

Inthissectionwewillshowthatnonlocalreductionsarisefrom scaling symmetries of integrable system of equations. A scaling symmetryofa systemofdifferentialequationsisthescale trans-formationwhichleavestheseequationsinvariant.Scaling symme-triesgroup is a subgroup ofthe symmetry groups ofdifferential

equations [39] and discrete symmetries are special cases of the scalingsymmetries[40].

(a) NLS system: Thissystemisgivenby

aqt

= −

1 2qxx

+

q 2_r

_,

₍₉₎ art

=

1 2rxx

−

q r 2

_,

₍₁₀₎

where a is anyconstant. This constant is the imaginary unit for theoriginalNLSsystembutwechangeitbyredefiningthet

vari-able.Wesearchforasymmetrytransformation suchthat theNLS systemisleftinvariant.Ingeneralwechoosethesymmetry trans-formationas

T1

: (

q

(

t

,

x

),

r

(

t

,

x

))

→ (

q

(

t

,

x

),

r

(

t

,

x

))

whereprimedsystemsatisfiesalsotheNLSsystem,i.e.,

aq_t

= −

1 2q  xx

+ (

q

)

2r

,

(11) ar_t

=

1 2r  xx

−

q

(

r

)

2

.

(12)

We shallconsiderthe realandcomplexdynamicalsystems sepa-rately.

(1) For therealcasethesymmetrytransformationthatweare in-terestedinisthescaletransformations

t

= β

t

,

x

=

α

x

,

q

=

γ

1q

+ δ

1r

,

r

=

γ

2r

+ δ

2q

,

(13) where

α

,

β,

γ

1

,

γ

2

,

δ

1,and

δ

2arerealconstants.Wehavetwo pos-siblecases:

(1.a) First typeofrealscalesymmetrytransformationis

t

= −

α

2_t

_,

_x

₌

_α

_x

_,

_q

_{= δ}

1r

,

r

=

1

δ

1

α

2

q

,

(14)

where

α

and

δ

1arearbitraryconstants.

(1.b) Second typeofrealscalesymmetrytransformationis

t

=

α

2t

,

x

=

α

x

,

q

=

γ

1q

,

r

=

1

γ

1

α

2

r

,

(15)

where

α

and

γ

1arearbitraryconstants.Thesetwoparameter scaletransformations map solutions tosolutions oftheNLS system.

Fromtheabovescalesymmetrytransformationwecanobtain dis-crete symmetry transformationsby letting

α

=

= ±

1. In partic-ularthefirsttypeproducesadiscretesymmetry transformationif

α

=

and

δ

1

=

k then

q

(

t

,

x

)

=

k r

(

−

t

,

x

),

r

(

t

,

x

)

=

k q

(

−

t

,

x

),

(16) where

2

₌

_k2

₌

_{1. A special discrete symmetry transformation} is obtained when we take q

=

q and r

=

r. Under this spe-cial discrete symmetry the transformations in(16) reduceto the well-known nonlocal reductions r

(

t

,

x

)

=

kq

(

−

t

,

x

)

and r

(

t

,

x

)

=

kq

(

−

t

,

−

x

)

[3],[4],[6],[10],[13],[14].

(2) For thecomplexdynamicalsystemsthescalesymmetry trans-formation

T2

: (¯

q

(

t

,

x

),

¯

r

(

t

,

x

))

→ (

q

(

t

,

x

),

r

(

t

,

x

))

takesthefollowingform

t

= β

t

,

x

=

α

x

,

q

=

γ

1q

¯

+ δ

1

¯

r

,

r

=

γ

2

¯

r

+ δ

2q

¯

,

(17) where

α

,

β,

γ

1

,

γ

2

,

δ

1,and

δ

2arerealconstants.Wehavetwo pos-siblecases:

(2.a) First typeofcomplexscalesymmetrytransformationis

t

= β

t

,

x

=

α

x

,

q

= δ

1r

¯

,

r

= δ

2q

¯

,

(18) with

¯

a

β

= −

a

α

2

,

δ

1

δ

2

α

2

=

1

.

(19) (2.b) Second typeofcomplexscalesymmetrytransformationis

t

= β

t

,

x

=

α

x

,

q

=

γ

1q

¯

,

r

=

γ

2r

¯

,

(20) with

¯

a

β

=

a

α

2

,

γ

1

γ

2

α

2

=

1

.

(21) Thesetwo parameterscaletransformations map alsosolutions to solutions of the NLS system. From these scale symmetry trans-formationsweobtaindiscretesymmetrytransformationbyletting

α

=

1

= ±

1,

β

=

2

= ±

1,

γ

1

=

γ

2

=

k

= ±

1.Inparticularthefirst typeproducesadiscretesymmetrytransformationoftheform

q

(

t

,

x

)

=

kr

¯



(

2t

,

1x

),

r

(

t

,

x

)

=

kq

¯



(

2t

,

1x

),

(22) where

2

1

=

22

=

k2

=

1 and a

¯

2

= −

a which follows from (19). Aspecialdiscrete symmetry transformationisobtainedwhen we take q

=

q and r

=

r. Under this special discrete symmetry the transformationsin(22) reducetothewell-knownnonlocal reduc-tionsr

(

t

,

x

)

=

kq

¯

(

−

t

,

x

)

witha

¯

= −

a,r

(

t

,

x

)

=

kq

¯

(

t

,

−

x

)

witha

¯

=

a,

andr

(

t

,

x

)

=

kq

¯

(

−

t

,

−

x

)

witha

¯

= −

a [1],[2],[4–9],[11–14]. Theexamples that we considerin therestof thepapershare similarrealandcomplexscalesymmetrytransformationsandthe associateddiscretesymmetrytransformations.Since weare inter-estedinnonlocalreductionsoftheintegrablesystemsofequations wewillpresentonlythefirsttyperealandcomplexdiscrete trans-formationsandthecorrespondingnonlocalreductions.

(b) MKdV system: Thissystemisgivenby

aqt

= −

1 4qxxx

+

3 2q r qx

,

(23) art

= −

1 4rxxx

+

3 2q r rx

.

(24)

Wewillwritethediscretesymmetry transformationsdirectly.We havetwo different cases: Let

(

q

,

r

)

and

(

q

,

r

)

satisfy the mKdV systemofequations(23) and(24).

Fortherealcasewehave

q

(

t

,

x

)

=

kr

(

2t

,

1x

),

r

(

t

,

x

)

=

kq

(

2t

,

1x

),

(25) where k2

=

1 and

1

2

=

1. When we take q

=

q and r

=

r weobtain thenonlocal reductionr

(

t

,

x

)

=

kq

(

−

t

,

−

x

)

[2–4], [13], [15–17].

Forthecomplexcasewehave

q

(

t

,

x

)

=

kr

¯



(

2t

,

1x

),

r

(

t

,

x

)

=

kq

¯



(

2t

,

1x

),

(26) wherea

¯

1

2

=

a andk2

=

1.Thesespecialdiscretetransformations producedifferentnonlocalreductionswhenq

=

q andr

=

r with

differentvaluesof

1

= ±

1 and

2

= ±

1; r

(

t

,

x

)

=

kq

¯

(

−

t

,

x

)

with

¯

a

= −

a, r

(

t

,

x

)

=

kq

¯

(

t

,

−

x

)

witha

¯

= −

a, andr

(

t

,

x

)

=

kq

¯

(

−

t

,

−

x

)

witha

¯

=

a [2–4],[13],[15],[18]. (c) SG system: Thissystemisgivenby

qxt

+

2s q

=

0

,

(27)

rxt

+

2s r

=

0

,

(28)

sx

+ (

q r

)

t

=

0

,

(29)

whereq

=

q

(

t

,

x

)

,r

=

r

(

t

,

x

)

,ands

=

s

(

t

,

x

)

.Wehavethefollowing twodiscretesymmetrytransformations.Fortherealcase,

q

(

t

,

x

)

=

kr

(

2t

,

1x

),

r

(

t

,

x

)

=

kq

(

2t

,

1x

),

s

(

t

,

x

)

=

s

(

2t

,

1x

),

(30) where

1

=

2

= ±

1 andk2

=

1.Ifwetakeq

=

q andr

=

r these special discrete transformations producethe nonlocal reductions:

r

(

t

,

x

)

=

kq

(

−

t

,

x

)

, r

(

t

,

x

)

=

kq

(

t

,

−

x

)

, and r

(

t

,

x

)

=

kq

(

−

t

,

−

x

)

[2–4],[19].

Forthecomplexcase,

q

(

t

,

x

)

=

k

¯

r

(

2t

,

1x

),

r

(

t

,

x

)

=

kq

¯



(

2t

,

1x

),

s

(

t

,

x

)

= ¯

s

(

2t

,

1x

),

(31) where

1

=

2

= ±

1 and k2

=

1. When q

=

q and r

=

r these special discrete transformations producethe nonlocal reductions:

r

(

t

,

x

)

=

kq

¯

(

−

t

,

x

)

,r

(

t

,

x

)

=

kq

¯

(

t

,

−

x

)

,andr

(

t

,

x

)

=

kq

¯

(

−

t

,

−

x

)

[4]. (d) DS system: Thissystemisgivenby

aqt

+

1 2

[

γ

2_q xx

+

qy y

] +

q2r

= φ

q

,

(32)

−

art

+

1 2

[

γ

2_r xx

+

ry y

] +

r2q

= φ

r

,

(33)

φ

xx

−

γ

2

φ

y y

=

2

(

qr

)

xx

,

(34) whereq

=

q

(

t

,

x

,

y

)

,r

=

r

(

t

,

x

,

y

)

,

φ

= φ(

t

,

x

,

y

)

,

γ

2

_{= ±}

_{1,anda is} aconstant. Wehavethefollowingdiscretesymmetry transforma-tions.Fortherealcase,

q

(

t

,

x

,

y

)

=

kr

(

1t

,

2x

,

3y

),

(35)

r

(

t

,

x

,

y

)

=

kq

(

1t

,

2x

,

3y

),

(36)

φ (

t

,

x

,

y

)

= φ



(

1t

,

2x

,

3y

),

(37) where

1

= −

1 andk2

=

1.Thesespecialdiscretetransformations producethenonlocalreductionswhenq

=

q andr

=

r with differ-entvaluesof

1

= −

1,

2

= ±

1,

3

= ±

1; r

(

t

,

x

,

y

)

=

kq

(

−

t

,

x

,

y

)

,

r

(

t

,

x

,

y

)

=

kq

(

−

t

,

−

x

,

y

)

,r

(

t

,

x

,

y

)

=

kq

(

−

t

,

x

,

−

y

)

,andr

(

t

,

x

,

y

)

=

kq

(

−

t

,

−

x

,

−

y

)

[3]. Forthecomplexcase,

q

(

t

,

x

,

y

)

=

k

¯

r

(

1t

,

2x

,

3y

),

(38)

r

(

t

,

x

,

y

)

=

kq

¯



(

1t

,

2x

,

3y

),

(39)

φ (

t

,

x

,

y

)

= ¯φ



(

1t

,

2x

,

3y

),

(40) where k2

₌

_1,

2

1

=

22

=

23

=

1, anda

¯

1

= −

a. We observethat these discrete transformations produce many different nonlocal reductions when q

=

q, r

=

r, and

φ



= φ

with different val-ues of

1

= ±

1,

2

= ±

1, and

3

= ±

1; r

(

t

,

x

,

y

)

=

kq

¯

(

−

t

,

x

,

y

)

,

r

(

t

,

x

,

y

)

=

kq

¯

(

−

t

,

−

x

,

y

)

, r

(

t

,

x

,

y

)

=

kq

¯

(

−

t

,

x

,

−

y

)

, r

(

t

,

x

,

y

)

=

kq

¯

(

−

t

,

−

x

,

−

y

)

with a

¯

=

a; r

(

t

,

x

,

y

)

=

kq

¯

(

t

,

−

x

,

y

)

, r

(

t

,

x

,

y

)

=

kq

¯

(

t

,

x

,

−

y

)

,r

(

t

,

x

,

y

)

=

kq

¯

(

t

,

−

x

,

−

y

)

witha

¯

= −

a [3],[20–24]. (e) Coupled NLS-derivative NLS system: Thissystem[41] isgiven by

aqt

=

iqxx

+

α

(

rq2

)

x

+

i

β

rq2

,

(41)

art

= −

irxx

+

α

(

rq2

)

x

−

i

β

r2q

,

(42) where

α

,

β

∈ R

,anda isanyconstant.Wehavethefollowing dis-cretesymmetrytransformations.Fortherealcase,

where

1

=

2

= −

1 and k2

=

1. When q

=

q and r

=

r, these discrete transformations produce the nonlocal reduction r

(

t

,

x

)

=

kq

(

−

t

,

−

x

)

[3].

Forthecomplexcase,

q

(

t

,

x

)

=

kr

¯



(

2t

,

1x

),

r

(

t

,

x

)

=

kq

¯



(

2t

,

1x

),

(44) where

1

=

1,a

¯

2

=

a, andk2

=

1. From thesediscrete transfor-mations we have different nonlocal reductions when q

=

q and r

=

r with different values of

1

= ±

1 and

2

= ±

1; r

(

t

,

x

)

=

kq

¯

(

−

t

,

x

)

witha

¯

=

a,r

(

t

,

x

)

=

kq

¯

(

t

,

−

x

)

witha

¯

= −

a, andr

(

t

,

x

)

=

kq

¯

(

−

t

,

−

x

)

witha

¯

= −

a.

(f) Loop-soliton system: Thissystem[41],[42] isgivenby

aqt

+

∂

2

∂

x2



_q x

(

1

−

rq

)

3/2



=

0

,

(45) art

+

∂

2

∂

x2



_r x

(

1

−

rq

)

3/2



=

0

.

(46)

Wehavethefollowingdiscretesymmetrytransformations. Fortherealcase,

q

(

t

,

x

)

=

kr

(

2t

,

1x

),

r

(

t

,

x

)

=

kq

(

2t

,

1x

),

(47) where

1

=

2

= −

1 and k2

=

1. When q

=

q and r

=

r, these discrete transformations produce the nonlocal reduction r

(

t

,

x

)

=

kq

(

−

t

,

−

x

)

[3].

Forthecomplexcase,

q

(

t

,

x

)

=

kr

¯



(

2t

,

1x

),

r

(

t

,

x

)

=

kq

¯



(

2t

,

1x

),

(48) wherea

¯

1

2

=

a, andk2

=

1.Thesediscrete transformations pro-duce different nonlocal reductions when q

=

q and r

=

r with

differentvaluesof

1

= ±

1 and

2

= ±

1;r

(

t

,

x

)

=

kq

¯

(

−

t

,

x

)

with

¯

a

= −

a, r

(

t

,

x

)

=

kq

¯

(

t

,

−

x

)

witha

¯

= −

a, and r

(

t

,

x

)

=

kq

¯

(

−

t

,

−

x

)

witha

¯

=

a.

(g) Hydrodynamic type of systems: Shallow water waves

Recently we studied the reductions in equations of hydrody-namictype[34] andobtainedseveralexamplesofnonlocalversion oftheseequations.Anexampleofequationsofhydrodynamictype istheshallowwaterwavessystem[43]

aqt

= (

q

+

r

)

qx

+

q rx

,

(49)

art

= (

q

+

r

)

rx

+

r qx

.

(50) Herea is anonzeroconstant. Thediscrete transformationswhich leavethissysteminvariantarefollowing.Fortherealcase,

r

(

t

,

x

)

=

k q

(

2t

,

1x

),

q

(

t

,

x

)

=

k r

(

2t

,

1x

),

(51) wherek

=

1

2.Forthecomplexcase

r

(

t

,

x

)

=

kq

¯



(

2t

,

1x

),

q

(

t

,

x

)

=

k

,

¯

r

(

2t

,

1x

),

(52) wherea k

¯

1

2

=

a.Inbothcasesk2

=

12

=

22

=

1 [34].

Ifweletq

=

q andr

=

r wegetthespecialdiscretesymmetry transformations which lead to the localand nonlocal reductions. Whenq andr arerealvariableswehaver

(

t

,

x

)

=

kq

(

2t

,

1x

)

then thereducedequationis

aqt

(

t

,

x

)

= (

q

(

t

,

x

)

+

kq

(

2t

,

1x

))

qx

(

t

,

x

)

+

kq

(

t

,

x

)

qx

(

2t

,

1x

),

(53) providedthatk

=

1

2 anda isreal.

Whenq andr arecomplexvariableswehaver

(

t

,

x

)

=

kq

¯

(

2t

.

1x

)

thenthereducedequationis

aqt

(

t

,

x

)

= (

q

(

t

,

x

)

+

kq

¯

(

2t

,

1x

))

qx

(

t

,

x

)

+

kq

(

t

,

x

)

q

¯

x

(

2t

,

1x

),

(54) providedthatak

¯

1

2

=

a [34].

(h) Fordy-Kulish equations

Let qα

(

t

,

x

)

and rα

(

t

,

x

)

be the complex dynamical variables where

α

=

1

,

2

,

· · · ,

N, then theFordy-Kulish (FK)integrable sys-temisgivenby[44]

aq_tα

=

qα_xx

+

Rαβγ−δqβqγrδ

,

(55)

−

arα_t

=

rα_xx

+

R−α_−β−γδrβrγqδ

,

(56)

where Rαβγ−δ

,

R−α_−β−γδ are thecurvature tensorsof a

Hermi-tiansymmetricspacewith

(

Rαβγ−δ

)



=

R−α_−β−γδ

,

(57)

anda isa complexnumber.Here weusethesummation conven-tion,i.e.,therepeatedindicesare summedupfrom1to N.These equationsare knownastheFKsystemwhichis integrableinthe sensethattheyareobtainedfromthezerocurvatureconditionofa connectiondefinedonaHermitiansymmetricspace.TheFK equa-tions(55) and(56) areinvariantunderthediscretetransformations

rα

(

t

,

x

)

=

kq

¯

α

(

1t

,

2x

),

qα

(

t

,

x

)

=

k

¯

rα

(

1t

,

2x

),

(58) wherek2

₌

2

1

=

2

2

=

1 anda

¯

1

= −

a.Ifweletrα

=

rα andqα

=

qα we obtainthe special discrete symmetry transformations and hence the nonlocal reductions rα

(

t

,

x

)

=

kqα

¯

(

1t

,

2x

)

[25]. Then thereducednonlocalFKequationsare

aqα_t

(

t

,

x

)

=

qα_xx

(

t

,

x

)

+

k Rαβγ−δqβ

(

t

,

x

)

qγ

(

t

,

x

)

q

¯

δ

(

1t

,

2x

).

(59) 4. Conclusion

In this work we showedthat thediscrete symmetries of sys-temsofintegrableequationsareimportantinfindingthenonlocal reductions. For this reason we started first with the scale sym-metry transformations of real and complex dynamical systems. Discrete symmetry transformations are special cases of the scale transformations. There are two differenttypes, calledasthe first and second types,ofdiscrete symmetry transformations both for realandcomplexdynamical variables.Usingthisfactwe canfind all discretesymmetrytransformations ofthesystemofequations. Amongthesediscretesymmetrytransformationsthefirsttypesare theoriginsofthenonlocalreductionsofthesesystems.Weshowed that a specialdiscrete symmetry transformation of thefirst type producesallthewell-knownnonlocalreductions.

Declaration of competing interest

Theauthorsdeclarethattheyhavenoknowncompeting finan-cialinterestsorpersonalrelationshipsthatcouldhaveappearedto influencetheworkreportedinthispaper.

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