Commun Nonlinear Sci Numer Simulat

ContentslistsavailableatScienceDirect

Commun Nonlinear Sci Numer Simulat

journalhomepage:www.elsevier.com/locate/cnsns

Research paper

Nonlocal hydrodynamic type of equations

Metin Gürses

^a

, Aslı Pekcan

^b,∗

, Kostyantyn Zheltukhin

^c

a Department of Mathematics, Faculty of Sciences, Bilkent University, Ankara 06800, Turkey

b Department of Mathematics, Hacettepe University, Ankara 06800, Turkey

c Department of Mathematics, Middle East Technical University, Ankara 06800, Turkey

a r t i c l e i n f o

Article history:

Received 27 June 2019 Revised 2 January 2020 Accepted 1 March 2020 Available online 2 March 2020 Keywords:

Hydrodynamic equations Lax representations Conserved quantities Nonlocal reductions

a b s t r a c t

Weshowthattheintegrableequations ofhydrodynamictypeadmitnonlocalreductions.

WefirstconstructsuchreductionsforageneralLaxequationandthengiveseveralexam- ples.Thereducednonlocalequationsareofhydrodynamictypeandintegrable.Theyadmit Laxrepresentationsandhencepossessinfinitelymanyconservedquantities.

© 2020 Elsevier B.V. All rights reserved.

1. Introduction

Letusconsiderthefollowingsystemofevolutionequationsin(¹+1)-dimension,

qⁱ_t=Fⁱ

(

^qk,r^k,q^k_x,r^k_x,q^k_xx,r^k_xx,...

)

, (1)

r_tⁱ=Gⁱ

(

^qk,rk,qkx,r^k_x,q^k_xx,r^k_xx,...

)

^{, (2)}

forall i,k=1,2,...,N.Here Fⁱ andGⁱ, (ⁱ=1,2,...,N)^{are functionsof thedynamical variables qi(x,t), ri(x,t), andtheir} partialderivativeswithrespecttox.IftheabovesystemhasaLaxpairthenitisanintegrablesystem.Therearelocaland nonlocalreductionsofthecoupledsystem(1)and(2).Thelocalreductionsaregivenby

rⁱ

(

^x,t

)

⁼

ρ

^qi

(

^x,t

)

^{, (3)}

and

rⁱ

(

x,t

)

=

ρ

^¯qi

(

x,t

)

, (4)

where

ρ

^{is a real constant and a bar over a letter denotes complex} conjugation. By these reductions the systems of Eqs.(1)and(2)reducetoonesystemforqⁱ’s

qⁱ_t=F˜ⁱ



q^j,q_x^j,q_xx^j,...



, i,j=1,2,...,N, (5)

∗ Corresponding author.

E-mail addresses: gurses@fen.bilkent.edu.tr (M. Gürses), aslipekcan@hacettepe.edu.tr (A. Pekcan), zheltukh@metu.edu.tr (K. Zheltukhin).

https://doi.org/10.1016/j.cnsns.2020.105242 1007-5704/© 2020 Elsevier B.V. All rights reserved.

so that thesystem (2) consistently reduces to the above system (5). Here corresponding to (3) we have F˜=F

|

^r=ρ^q and F˜=F

|

r=ρ^¯qcorresponding to (4).There are also nonlocal reductions introduced by Ablowitz and Musslimani[1–3] given as

rⁱ

(

^x,t

)

⁼

ρ

^qi

( ε

^1x,

ε

^2t

)

^{, (6)}

and

rⁱ

(

^x,t

)

=

ρ

^¯qi

( ε

1x,

ε

2t

)

, (7)

for i=1,2,...,N.Here

ρ

^{is a real constant and}

ε

²1=

ε

2²=1.If

ε

1=

ε

2=1the above reductions turnto be local reduc- tions.When(

ε

1,

ε

2)=(¹,−1),(−1,1),(−1,−1),wereducethesystem(1)and(2)bythesenonlocalreductionstononlocal timereflectionsymmetric(T-symmetric),spacereflectionsymmetric(S-symmetric),orspace-timereflectionsymmetric(ST- symmetric)differentialequations.

Thenonlocal reductionsofsystemsofintegrableequationswere firstconsistentlyappliedto thenonlinearSchrödinger system,whichledtothenonlocalnonlinearSchrödingerequation(nNLS)[1–3].Ithasbeenlatershownthattherearemany othersystemswheretheconsistentnonlocalreductionsarepossible[4–29].Forarecentreviewofthissubjectsee[27].See also[30]forthediscussionofsuperpositionofnonlocalintegrableequationsand[31]fortheoriginofnonlocalreductions.

One wayof obtaininga systemofintegrableequations,inGelfand-Dikiiformalism [32], istoconstructa Laxoperator onsomeLiealgebra.Theexamplesofsuchalgebrasarethematrixalgebra,algebraofdifferentialoperators,andalgebraof Laurentseries(see[33–35]and references there in).In most casesthe Lax operator is polynomial.It is a polynomial of the spectralparameterinthematrixalgebra,theoperatorDxinthealgebraofdifferentialoperators,andanauxiliaryvariablep (momentum)inthecaseofthealgebraofLaurentseries.

WhenwewritetheLaxequationonthealgebraofLaurentserieswederiveequationsofthehydrodynamictype,thatis asystemoffirstorderquasilinearpartialdifferentialequations

unt=

k

m=1

hmn

(

^u

)

^{umx, n=1,2,...,k, (8)}

where u=(^u1,. . .,u_k)^{. The derived systems are} integrable. This means they admit recursion operators and multi- Hamiltonianrepresentation(see[33,36–38]).

Fora hydrodynamictype systemwealso havelocalreductions(3),(4),andnonlocal reductions(6),(7)consistent for

ρ

²=1.Localreductionsforthehydrodynamictypesystemswereconsideredin[36].Inthisworkweaddresstotheproblem ofnonlocalreductionsfortheLaxequationsonthealgebraofLaurentseries.

Thelayoutofthepaperisasfollows.InSection2we giveashortreviewonthehydrodynamictypeof(dispersionless) equations andtheir Lax representations.In Section 3 we give nonlocal reductionsforreal andcomplex valuedfields for N=2.Weobtain someexplicitexamples ofnonlocalhydrodynamictypeofequationswithsome conservedquantities.In Section4wediscussthenonlocalreductionsingeneralandgiveexamplesforhighervaluesofN.

2. Laxequations

In thissection we describethe algebra ofLaurentseries,introduce necessary definitionsto write aLax equation, and givesomeexamples.Inlatersectionswegivenonlocalreductionsoftheseequations.

WeconsideranalgebraA,givenby A=

 α

^=∞

−∞

α

ⁿ

(

x,t

)

^{pn :}

α

ⁿ

(

x,t

)

∈Y



, (9)

whereYiseitherC^∞(S¹)-thespaceofperiodicfunctionsorS(R)^{-thespaceofsmooth} asymptotically decreasingfunctions onR.APoissonbracketonA,isgivenby

{

^f,g

}

k=p^k

 ∂

^f

∂

^p

∂

^g

∂

^x−

∂

^f

∂

^x

∂

^g

∂

^p



, f,g∈A, (10)

wherek∈Z.

OnthealgebraAwecandefineatracefunctional tr_k

α

=

I

Res_k

α

^dx,

α

∈A, (11)

where Res_k

α

=

α

k−1(^x,t)^{. The set Iis either S1 forthe spaceof periodic functionsor} Rfor the spaceof asymptotically decayingfunctions.Usingthetracefunctionalwedefineanon-degenerate,ad-invariant,symmetricpairing

( α

,

β )

= trk

α

·

β

,

α

,

β

∈A. (12)

ThealgebraAadmitsadecompositionintosub-algebras,closedwithrespecttothePoissonbracket,

A=A_≥−k+1A_<−k+1, (13)

where

A_≥−k+1=

α

≥−k+1=

∞

−k+1

α

ⁿ

(

x,t

)

^pn:

α

ⁿ

(

x,t

)

∈Y



(14)

and

A_<−k+1=

α

<−k+1=^−k

−∞

α

ⁿ

(

x,t

)

^pn:

α

ⁿ

(

x,t

)

∈Y



. (15)

So,wecandefiner-matrixmappingonA(see[35]andreferencestherein) R_k=1

2

 

≥−k+1−



<−k+1



,

where_≥−k+1^and_<−k+1^areprojectionsonthesub-algebrasA_≥−k+1 andA_<−k+1respectively.

NowforaLaxoperator L=p^N−1+

N−2

i=−1

pⁱSi

(

x,t

)

, N∈N, (16)

wecanconsiderahierarchyoftheLaxequations(see[36])

∂

^L

∂

^{tn =}



R_k

(

^{LN−1m +n}

)

,L

k=



L

m N−1+n

≥−k+1,L



k, n=1,2,..., (17)

wherem=0,1,2,...,(^N− 2) ^{isfixed. This is an integrable hierarchythat is the Lax equationsadmit} multi-Hamiltonian representation(see[33]).Inparticulartheconservedquantitiesforthehierarchyaregivenby

Qn= trkL^{N−1m +n}, n=1,2,.... (18)

Wenote that to considernonlocal reductionsitismore convenientto write theEq.(17)interms ofzerosof theLax operator.Thatisweset

L= 1 p

N

j=1

(

^p− uj

)

. (19)

LetusgivesomeexamplesoftheLaxequationsandtheirconservedquantities.First,weconsiderexamplesofLaxequa- tionscorrespondingtotheLaxoperator

L= 1

p

(

^p− u

)(

^p−

v ₎

. (20)

Example1. WetakethealgebraAwithk=0andconsidertheLaxEq.(17)withaLaxoperator(20).Forn=2,weobtain theshallowwaterwavessystem

1

2ut2=

(

^u+

v ₎

ux+u

v

x,1

2

v

t2=

(

^u+

v ₎ v

x+

v

ux. (21)

Taking n=3,4,..., we can have generalized symmetriesof the shallowwater waves system. Forinstance, the first two symmetriesare

1

3ut3=

(

^u2+4u

v

+

v

²

₎

ux+

(

^2u2+2u

v ₎ v

x,1

3

v

t3=

(

²

v

²+2u

v ₎

ux+

( v

²+4u

v

+u²

) v

x, (22) and

1

4ut4 =



u³+9u²

v

+9u

v

²+

v

³



ux+



3u³+9u²

v

+3u

v

²

 v

x, 1

4

v

t4 =



3

v

³+9

v

²u+3

v

u²



ux+



v

³+9

v

²u+9

v

u²+u³

 v

x.

(23)

Theabovesystemsadmitinfinitelymanyconservedquantities.Thefirstthreeoftheconservedquantitiesare Q2=

I

(

^u2

v

+u

v

²

)

dx, (24)

Q3=

I

(

^u3

v

+3u²

v

²+u

v

³

)

dx, (25)

Q4=

I

(

^u4

v

+6u³

v

²+6u²

v

³+6u

v

⁴

)

dx. (26)

Example2. WetakethealgebraAwithk=1andconsidertheLaxEq.(17)withaLaxoperator(20).Forn=1we obtain theTodasystem

ut1=u

v

x,

v

t1=

v

ux. (27)

Takingn=2,3,...,wecanhavegeneralizedsymmetriesoftheTodasystem.TheTodasystemadmitsinfinitelymanycon- servedquantities.Thefirstthreeoftheconservedquantitiesare

Q1=

I

(

^u+

v )

dx, (28)

Q2=

I

(

^u2+4u

v

+

v

²

)

dx, (29)

Q3=

I

(

^u3+9u2

v

+9u

v

²+

v

³

₎

dx. (30)

LetusgiveexamplesoftheLaxequationwithmoregeneralLaxoperator.

Example3. WetakethealgebraAwithk=0andconsidertheLaxEq.(17)withaLaxoperator L=1

p

(

^p− u

)(

^p−

v ₎₍

p− w

)

, (31)

andm=1.Forn=1weobtainthefollowingsystem 8

3ut1 =



−u²+

v

²+w²+4uv+4uw+6vw



ux+



2u²+2uv+6uw

 v

x+



2u²+6uv+2uw



wx, 8

3

v

t1 =



2

v

²+2uv+6vw



ux+



−

v

²+u²+w²+4uv+4vw+6vw

 v

x+



2

v

²+6uv+2vw



wx, 8

3wt1 =



2w²+6wv+2uw



ux+



2w²+2wv+6uw

 v

x+



−w²+

v

²+u²+4wv+4uw+6uv



wx.

(32)

Taking n=2,3,..., we can have generalizedsymmetries ofthe above system. The above system admitsinfinitely many conservedquantities.Thefirsttwooftheconservedquantitiesare

Q1=

I

(

^u2+

v

²+w²− 2u

v

− 2uw− 2

v

w

)

dx, (33)

Q2 =

I

(

^u4+

v

⁴+w⁴− 4

(

^u3

v

− u³w− u

v

³−

v

³w− uw³−

v

w³

)

+6

(

^u2

v

²+u²w²+6

v

²w²

)

+36

(

^u2

v

w+u

v

²w+u

v

w²

))

dx, (34)

Example4. Forsimplicity,wetakethealgebraAwithk=0andconsidertheLaxEq.(17)withaLaxoperator L=1

p

(

^{p− u}

) (

^{p− u1}

) (

^p−

v _{) (}

p−

v

1

)

^{, (35)}

andm=0.Forn=1weobtainthefollowingsystem

ut1 =

(

^u1

v

+u1

v

1+

vv

1

)

^ux+

(

^u

v

+u

v

1

)

^u1x+

(

^uu1+u

v

1

) v

x+

(

^uu1+u

v ₎ v

1x, u_1t1=

(

^u

v

+u

v

1+

vv

1

)

^u1x+

(

^u1

v

+u₁

v

1

)

^ux+

(

^uu1+u₁

v

1

) v

x+

(

^u1u+u₁

v ₎ v

1x,

v

t1 =

(

^uu1+u

v

1+u1

v

1

) v

x+

(

^u1

v

+

vv

1

)

^ux+

(

^u

v

+

vv

1

)

^u1x+

(

^u

v

+u1

v ) v

1x,

v

1t1 =

(

^uu1+u

v

+u1

v ) v

1x+

(

^u1

v

1+

vv

1

)

^ux+

(

^u

v

1+

vv

1

)

^u1x+

(

^u

v

1+u1

v

1

) v

x. (36) Taking n=2,3,..., we can have generalizedsymmetries ofthe above system. The above system admitsinfinitely many conservedquantities.Thefirsttwooftheconservedquantitiesare

Q1=

I

uu₁

vv

1dx, (37)

Q2=

I

(

^u2u21

v

²

v

1+u²u²₁

vv

²₁+u²u1

v

²

v

²₁+uu²₁

v

²

v

²₁

₎

dx. (38)

3. NonlocalreductionsfortwocomponentLaxoperator

TostudynonlocalreductionsoftheLaxEq.(17),westartwiththecaseoftwocomponentsystems.Thatiswestudythe hierarchy

∂

^L

∂

^tn =



Lⁿ_≥−k+1; L

k, n=1,2,... (39)

withtheLaxoperator(20).

3.1. Nonlocalreductionsforrealvalueddependentvariables

Inthissectionweassumethatourdependentvariablesarerealvaluedandconsiderreductionoftheform

v ₍

x,t

)

=

ρ

^u

( ε

1x,

ε

2t

)

=

ρ

^uε. (40)

ThereducedequationistheLaxEq.(39)withtheLaxoperator L= 1

p

(

^p− u

)(

^p−

ρ

^uε

)

. (41)

Tofindadmissiblereductionsweintroducethefollowingchangeofvariables

˜

x=

ε

1x, t˜n=

ε

2tn, p˜=

ρ

p. (42)

InnewvariablestheLaxoperatorbecomes L=

ρ

p˜

( ρ

^p˜− u^ε

)( ρ

^p˜−

ρ

^u

)

=

ρ

^L˜, (43)

whereL˜= 1

˜

p(^p˜− u)(^p˜−

ρ

^uε)^{.TheLaxEq.(39)takestheform}

ε

2

∂

∂

^t˜n

⁽ ρ

^L˜

)

⁼

( ρ

^p˜

)

^k



ρ ∂ ∂

^p˜

⁽ ρ

^nL˜n_≥−k+1

) ε

1

∂

∂

^x˜

⁽ ρ

^L˜

)

⁻

ε

1

∂

∂

^x˜

⁽ ρ

^nL˜n_≥−k+1

) ρ ∂ ∂

^p˜

⁽ ρ

^L˜

)



(44)

or

ρ

^k+n+1

ε

²

ε

¹

∂

∂

^t˜n

⁽

^˜L

⁾

^=p˜k

 ∂

∂

^{p˜L˜n≥−k+1}

∂

∂

^x˜L˜−

∂

∂

^{x˜L˜n≥−k+1}

∂

∂

^p˜L˜



. (45)

It is easy to see that if

ρ

^k+n+1

ε

2

ε

1=1 then the hierarchy(39) withLax operator (41) is invariant under thechange of variablesandwehaveaconsistentnonlocalreduction.

Since

ρ

²=1,wehaveonecondition

ρ

^k

ε

2

ε

1=1forall oddnandanothercondition

ρ

^k+1

ε

2

ε

1=1forallevenn.Thus, toconsidernonlocalreductionofthewholehierarchy(39)wehavetosplititintotwohierarchies

∂

^L

∂

^t2n =



L²_≥−kⁿ ₊₁; L

k, n=1,2,... (46)

and

∂

^L

∂

^t2n−1 =



L²_≥−kⁿ⁻¹₊₁; L

k, n=1,2,.... (47)

Taking

ρ

^,

ε

1,and

ε

2satisfying

ρ

^k+1

ε

²

ε

^{1=1, (48)}

weobtainnonlocalreductionsforthehierarchy(46).Notethattheseequationsadmitthefollowingconservedquantities

Q2n= tr_kL²ⁿ, n=1,2,..., (49)

whereLisgivenby(41). Taking

ρ

^,

ε

1,and

ε

2satisfying

ρ

^k

ε

2

ε

1=1, (50)

weobtainnonlocalreductionsforthehierarchy(47).Notethattheseequationsadmitthefollowingconservedquantities

Q2n−1= tr_kL²ⁿ⁻¹, n=1,2,..., (51)

whereLisgivenby(41).

Letusgivesomeexamples.Asafirstexampleweconsidertheshallowwaterwavessystem. Notethat fromnowonin allexamplesinthetextwelettn=t foralln.

Example5. Tohavenonlocalreductionsoftheshallowwaterwavessystem(21),weneedtoworkwiththehierarchy(46). Taking

ρ

^,

ε

1,and

ε

2satisfying(48)weobtainpossiblereductions.

a. S-symmetricreduction,

ρ

=−1,

ε

1=−1,

ε

2=1,is 1

2ut

(

^x,t

)

⁼

(

^u

(

^x,t

)

^{− u}

(

^−x,t

) )

^ux

(

^x,t

)

^{− u}

(

^x,t

)

^ux

(

^−x,t

)

^{. (52)}

Ithasinfinitelymanysymmetries,withthefirstsymmetry,thereductionof(23),being

1

4ut

(

^x,t

)

⁼



u³

(

^x,t

)

^{− 9u2}

(

^x,t

)

^u

(

^−x,t

)

^+9u

(

^x,t

)

^u2

(

^−x,t

)

^{− u3}

(

^−x,t

) 

ux

(

^x,t

)

−



3u³

(

^x,t

)

^{− 9u2}

(

^x,t

)

^u

(

^−x,t

)

^+3u

(

^x,t

)

^u2

(

^−x,t

) 

ux

(

^−x,t

)

^{. (53)}

Theaboveequationshaveinfinitelymanyconservedquantities.Thefirsttwoconservedquantitiesare Q2=

I

(

−u²

(

x,t

)

^u

(

−x,t

)

+u

(

x,t

)

^u2

(

−x,t

))

dx, (54)

Q4=

I

(

^−u4

(

^x,t

)

^u

(

^−x,t

)

^+6u3

(

^x,t

)

^u2

(

^−x,t

)

^{− 6u2}

(

^x,t

)

^u3

(

^−x,t

)

^+u

(

^x,t

)

^u4

(

^−x,t

))

^{dx. (55)}

b. T-symmetricreduction,

ρ

=−1,

ε

1=1,

ε

2=−1,is 1

2ut

(

^x,t

)

⁼

(

^u

(

^x,t

)

^{− u}

(

^x,−t

) )

^ux

(

^x,t

)

^{− u}

(

^x,t

)

^ux

(

^x,−t

)

^{. (56)}

Ithasinfinitelymanysymmetries,withthefirstsymmetry,thereductionof(23),being 1

4ut

(

^x,t

)

=



u³

(

^x,t

)

− 9u²

(

^x,t

)

^u

(

^x,−t

)

+9u

(

^x,t

)

^u2

(

^x,−t

)

− u³

(

^x,−t

) 

ux

(

^x,t

)

−



3u³

(

^x,t

)

^{− 9u2}

(

^x,t

)

^u

(

^x,−t

)

^+3u

(

^x,t

)

^u2

(

^x,−t

) 

ux

(

^x,−t

)

^{. (57)}

Theaboveequationshaveinfinitelymanyconservedquantities.Thefirsttwoconservedquantitiesare Q2=

I

(

−u²

(

^x,t

)

^u

(

^x,−t

)

+u

(

^x,t

)

^u2

(

^x,−t

))

^dx, (58)

Q4=

I

(

−u⁴

(

x,t

)

^u

(

x,−t

)

+6u³

(

x,t

)

^u

(

x,−t

)

²− 6u²

(

x,t

)

^u

(

x,−t

)

³+6u

(

x,t

)

^u4

(

x,−t

))

dx. (59) c. ST-symmetricreduction,

ρ

=1,

ε

1=−1,

ε

2=−1,is

1

2ut

(

^x,t

)

⁼

(

^u

(

^x,t

)

^+u

(

^−x,−t

) )

^ux

(

^x,t

)

^+u

(

^x,t

)

^ux

(

^−x,−t

)

^{. (60)}

Ithasinfinitelymanysymmetrieswiththefirstsymmetry,thereductionof(23),being 1

4ut

(

^x,t

)

⁼



u³

(

^x,t

)

^+9u2

(

^x,t

)

^u

(

^−x,−t

)

^+9u

(

^x,t

)

^u2

(

^−x,−t

)

^+u3

(

^−x,−t

) 

ux

(

^x,t

)

+



3u³

(

x,t

)

+9u²

(

x,t

)

^u

(

−x,−t

)

+3u

(

x,t

)

^u2

(

−x,−t

) 

ux

(

−x,−t

)

. (61)

Theaboveequationshaveinfinitelymanyconservedquantities.Thefirsttwoconservedquantitiesare Q2=

I

(

^u2

(

x,t

)

^u

(

−x,−t

)

+u²

(

x,t

)

^u

(

−x,−t

))

dx, (62)

Q4=

I

(

^u4

(

x,t

)

^u

(

−x,−t

)

+6u³

(

x,t

)

^u

(

−x,−t

)

²+6u²

(

x,t

)

^u3

(

−x,−t

)

+u

(

x,t

)

^u4

(

−x,−t

))

dx. (63) Wecanalsoconsiderthereductionsfortheoddsymmetriesoftheshallowwaterwavesequations.

Example6. Tohavenonlocalreductionsofthesystem(22),weneedtoworkwiththehierarchy(47).Taking

ρ

^,

ε

1,and

ε

2, satisfying(50)weobtainonepossiblereduction.

ST-symmetricreduction,

ρ

=1,

ε

1=−1,

ε

2=−1,is 1

3ut

(

^x,t

)

⁼



u

(

^x,t

)

^2+4u

(

^x,t

)

^u

(

^−x,−t

)

^+u2

(

^−x,−t

) 

ux

(

^x,t

)

⁺



2u²

(

^x,t

)

^+2u

(

^x,t

)

^u

(

^−x,−t

) 

ux

(

^−x,−t

)

^{. (64)}

Theabovesystemhasinfinitelymanysymmetriesandconservedquantities.Thefirsttwoconservedquantitiesare Q3=

I

(

^u3

(

^x,t

)

^u

(

−x,−t

)

+3u²

(

^x,t

)

^u2

(

−x,−t

)

+u

(

^x,t

)

^u

(

−x,−t

)

³

)

^dx, (65) Q5 =

I

(

^u5

(

^x,t

)

^u

(

^−x,−t

)

^+10u4

(

^x,t

)

^u2

(

^−x,−t

)

^+20u3

(

^x,t

)

^u3

(

^−x,−t

)

+10u²

(

^x,t

)

^u4

(

−x,−t

)

+u

(

^x,t

)

^u5

(

−x,−t

))

^dx. (66) LetusalsoconsiderthereductionsoftheTodasystem.

Example7. TohavenonlocalreductionsoftheTodasystem(27),weneedtoworkwiththehierarchy(47).Taking

ρ

^,

ε

1,and

ε

2 satisfying(50)wecanobtainthefollowingreductions:S-symmetricreduction,

ρ

=−1,

ε

1=−1,

ε

2=1;T-symmetricre- duction,

ρ

=−1,

ε

1=1,

ε

2=−1;ST-symmetricreduction,

ρ

=1,

ε

1=−1,

ε

2=−1.Forinstance,theST-symmetricreduction oftheTodasystemis

ut

(

^x,t

)

^=u

(

^x,t

)

^ux

(

^−x,−t

)

^{. (67)}

Ithasinfinitelymanysymmetries,withthefirstsymmetry 1

3ut

(

x,t

)

=2u

(

x,t

) 

u

(

x,t

)

^u

(

−x,−t

)

+u²

(

−x,−t

) 

ux

(

x,t

)

+u

(

^x,t

) 

u²

(

^x,t

)

^+4u

(

^x,t

)

^u

(

^−x,−t

)

^+u2

(

^x,t

) 

ux

(

^−x,−t

)

^{, (68)}

whichis the ST-symmetricreduction of thesecond equation inthe hierarchy (39). It alsohas infinitelymany conserved quantities.Thefirsttwoconservedquantitiesare

Q1=

I

(

^u

(

x,t

)

+u

(

−x,−t

))

dx, (69)

Q3=

I

(

^u3

(

x,t

)

+9u²

(

x,t

)

^u

(

−x,−t

)

+9u

(

x,t

)

^u2

(

−x,−t

)

− u³

(

−x,−t

))

dx. (70) Remark3.1. Weconsideredthereductionsforthewholehierarchy(39).Wecanalsoconsidernonlocalreductionsjustfor oneequationfromthehierarchy.Forinstance,takingk=0andn=2in(39)wegetthefollowingequations:

1

2ut=

(

^u+

v ₎

ux+u

v

x, (71)

1

2

v

t=

(

^u+

v ₎ v

x+

v

ux. (72)

Letusconsider nonlocalreductionsoftheseequationsonly. Assumethat

v

₍x,t)=

ρ

^u(

ε

1x,

ε

2t)=

ρ

^uε,where

ε

²1=

ε

²2=1, and

ρ

^{isarealconstant,thentheabovesystemtakesform}

1

2

ρ

^ut=

⁽

^uε+

1

ρ

^u

⁾

^{ux+uuεx, (73)}

1

2

ε

¹

ε

^2ut=

(

^uε+

ρ

^u

)

^ux+uuεx. (74)

Forconsistency,we musthave

ρ

=

ε

1

ε

2 which isthesameconditionasequality(48).Therefore workingwiththe whole hierarchyweobtainallpossiblecases.

3.2.Nonlocalreductionsforcomplexvalueddependentvariables

Inthissectionweassumethatalldependentvariablesarecomplexvaluedfunctions.Tostudyreductionsofsuchequa- tionsitisconvenienttointroduceaconstantintheLaxequation (39)bysettingnewtimet^=at,a∈C.Soweconsidera hierarchy

a

∂

^L

∂

^tn =



Lⁿ_≥−k+1; L

k, n=1,2,.... (75)

Notethatafterthechangeofthetimevariableweshalluseagainttodenotenewtime.

Thereductionisoftheform

v (

x,t

)

=

ρ

^u¯

( ε

^1x,

ε

^2t

)

=

ρ

^{u¯ε. (76)}

Tofindadmissiblereductionsweusethefollowingchangeofvariables

˜

x=

ε

1x, t˜n=

ε

2tn, p˜=

ρ

^p. (77)

InnewvariablestheLaxoperatorbecomes L=

ρ

˜

p

( ρ

^p˜− u^ε

)( ρ

^p˜−

ρ

^u¯

)

=

ρ

^¯L, (78)

whereL^=1

˜

p(^p˜− u)(^p˜−

ρ

^u¯ε)^{andtheLaxEq.(39)takestheform} a

ε

²

∂

∂

^t˜n

⁽ ρ

^¯L

)

=

( ρ

^p˜

)

^k



ρ ∂ ∂

^p˜

⁽ ρ

ⁿ

(

^¯L

)

ⁿ_≥−k+1

) ε

¹

∂

∂

^x˜

⁽ ρ

^¯L

)

−

ε

¹

∂

∂

^x˜

⁽ ρ

ⁿ

(

^¯L

)

ⁿ_≥−k+1

) ρ ∂ ∂

^p˜

⁽ ρ

^¯L

)



. (79)

Takingcomplexconjugatesofbothsidesintheaboveequalitywecanget a

ρ

^k+n+1

ε

2

ε

1

∂

∂

^t˜n

(

^L

)

=p˜^k

 ∂

∂

^p˜

⁽

^L

⁾

^n≥−k+1

∂

∂

^x˜L−

∂

∂

^x˜

⁽

^L

⁾

^n≥−k+1

∂

∂

^p˜L



. (80)

Ifa

ρ

^k+n+1

ε

2

ε

1=a then the hierarchy(75)is invariant under thechange ofvariables andwe havea consistent nonlocal reduction.

Since

ρ

²=1wehaveoneconditiona

ρ

^k

ε

2

ε

1=aforalloddnandanotherconditiona

ρ

^k+1

ε

2

ε

1=aforallevenn.Thus, toconsiderlocalreductionofthehierarchy(75)wehavetosplititintotwohierarchies(46)and(47)asintherealcase.

Letusgiveoneexampleofthereductionforcomplexvalueddependentvariables.

Example8. Wetakek=0andconsiderthehierarchy(75)withneven.Thefirstsystemcorrespondington=2is aut = 2

(

^u+

v ₎

ux+2u

v

x,

avt = 2

(

^u+

v ₎ v

x+2

v

ux. (81)

Taking

ρ

^,

ε

1,and

ε

2 satisfyinga

ρε

2

ε

1=aweobtainpossiblereductions.

a. S-symmetricreduction,

ε

1=−1,

ε

2=1,is a

2ut

(

^x,t

)

⁼

(

^u

(

^x,t

)

⁺

ρ

^u¯

(

^−x,t

))

^ux

(

^x,t

)

⁺

ρ

^u

(

^x,t

)

^u¯x

(

^−x,t

)

^{, (82)}

where we can take

ρ

=1 and the constant a equals to a pure imaginary number or

ρ

=−1 and the constant a equalstoarealnumber.Theaboveequation hasinfinitelymanysymmetriesandconservedquantities.Thefirsttwo conservedquantitiesare

Q2=

I

(

^−u2

(

^x,t

) ρ

^u¯

(

^−x,t

)

^+u

(

^x,t

) ρ

^u¯2

(

^−x,t

))

^{dx, (83)}

Q4=

I

(

^−u4

(

^x,t

) ρ

^u¯

(

^−x,t

)

^+6u3

(

^x,t

) ρ

^u¯2

(

^−x,t

)

^{− 6u2}

(

^x,t

) ρ

^u¯3

(

^−x,t

)

^+u

(

^x,t

) ρ

^u¯4

(

^−x,t

))

^{dx. (84)}

b. T-symmetricreduction,

ε

1=1,

ε

2=−1,is a

2ut

(

x,t

)

=

(

^u

(

x,t

)

+

ρ

^u¯

(

x,−t

))

^ux

(

x,t

)

+

ρ

^u

(

x,t

)

^u¯x

(

x,−t

)

, (85) where we can take

ρ

=1 and the constant a equals to a pure imaginary number or

ρ

=−1 and the constant a equalstoarealnumber.Theaboveequation hasinfinitelymanysymmetriesandconservedquantities.Thefirsttwo conservedquantitiesare

Q2=

I

(

−u²

(

x,t

) ρ

^u¯

(

x,−t

)

+u

(

x,t

) ρ

^u¯2

(

x,−t

))

dx, (86)

Q4=

I

(

−u⁴

(

x,t

) ρ

^u¯

(

x,−t

)

+6u³

(

x,t

) ρ

^u¯2

(

x,−t

)

− 6u²

(

x,t

) ρ

^u¯3

(

x,−t

)

+u

(

x,t

) ρ

^u¯4

(

x,−t

))

dx. (87) c. ST-symmetricreduction,

ε

1=−1,

ε

2=−1,is

a

2ut

(

^x,t

)

=

(

^u

(

^x,t

)

+

ρ

^u¯

(

−x,−t

))

^ux

(

^x,t

)

+

ρ

^u

(

^x,t

)

^u¯x

(

−x,−t

)

, (88) where we can take

ρ

=1 and the constant a equals to a real number or

ρ

=−1 andthe constant a equals to a pureimaginarynumber.Theaboveequation hasinfinitelymanysymmetriesandconservedquantities.Thefirsttwo conservedquantitiesare

Q2=

I

(

−u²

(

^x,t

) ρ

^u¯

(

−x,−t

)

+u

(

^x,t

) ρ

^u¯2

(

−x,−t

))

^dx, (89)

Q4=

I

(

−u⁴

(

x,t

) ρ

^u¯

(

−x,−t

)

+6u³

(

x,t

) ρ

^u¯2

(

−x,−t

)

− 6u²

(

x,t

) ρ

^u¯3

(

−x,−t

)

+u

(

x,t

) ρ

^u¯4

(

−x,−t

))

dx. (90) 4. ReductionswithageneralLaxoperator

TostudynonlocalreductionsoftheLaxEq.(17)withthegeneralLaxoperator(19),wecandividethevariablesu_iinto pairsandsetvariablesthatdonothaveapairtozero.Foreachpairofvariablesweconsiderarelationsimilarto(40)and proceedasbefore.Asanexampleletusconsidertwospecialcases.

4.1. ThreecomponentLaxoperator

InthissectionwetaketheLaxoperator L=1

p

(

^p− u

)(

^p−

v ₎₍

p− w

)

, (91)

andconsiderthehierarchy(17)withm=1.Thuswehave

∂

^L

∂

^{tn =}



L_≥−k¹²⁺ⁿ₊₁; L



k, n=1,2,.... (92)