by Hirota Method Nonlocal modiﬁed KdV equations and their soliton solutions Commun Nonlinear Sci Numer Simulat

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Commun Nonlinear Sci Numer Simulat

journalhomepage:www.elsevier.com/locate/cnsns

Research paper

Nonlocal modiﬁed KdV equations and their soliton solutions by Hirota Method

Metin Gürses

^a

, Aslı Pekcan

^b^,^∗

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

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

a r t i c l e i n f o

Article history:

Received 24 March 2018 Accepted 9 July 2018 Available online 20 July 2018 Keywords:

Ablowitz–Musslimani reduction Nonlocal mKdV equations Hirota bilinear form Soliton solutions

a b s t r a c t

WestudythenonlocalmodifiedKorteweg–deVries(mKdV)equationsobtainedfromAKNS schemebyAblowitz–Musslimanitypenonlocalreductions.Wefirstfindsolitonsolutions of the coupled mKdV system by using the Hirota direct method. Then by using the Ablowitz–Musslimanireductionformulas,wefindone-,two-,andthree-solitonsolutions ofnonlocal mKdVandnonlocal complexmKdVequations.Thesolitonsolutionsofthese equationsareoftwotypes.Wegiveone-solitonsolutionsofbothtypesandpresentonly firsttypeoftwo-andthree-solitonsolutions.Weillustrateoursolutionsbyplottingtheir graphsforparticularvaluesoftheparameters.

1. Introduction

Nonlocalintegrableequationsstudied so farare of integro-differential equation type, such asBenjamin-Onoequation.

Recently[1]AblowitzandMusslimanihaveintroducedanewtype ofnonlocalintegrableequations.Inthesenewtypesof nonlocalequations,inadditiontothetermsatthespace-timepoint(t,x),therearetermsatthemirrorimagepoint(^t,−x)^. All such new integrableequations seem to be obtained by a reduction from an integrable system ofcoupled integrable equations.For instance when the Lax pair is a cubic polynomial of the spectral parameter we obtain coupled modiﬁed Korteweg–deVriessystemofequationsfromtheAKNSformalism[6].Theseequationsaregivenby

aqt=−1 4qxxx+3

2rqqx, (1)

art=−1 4rxxx+3

2rqrx, (2)

whereq(t,x)andr(t,x) areingeneralcomplexdynamicalvariables,a isaconstant.We calltheabove systemofcoupled equationsasnonlinearmodiﬁedKorteweg–deVriessystem(mKdVsystem).Wehavetwo differentlocal(standard) reduc- tionsofthissystem:

a

)

^r

(

^t,x

)

=k¯q

(

^t,x

)

, (3)

∗ Corresponding author.

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

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b

)

^r

(

t,x

)

=kq

(

t,x

)

, (4) where k is a real constant and ¯q is the complex conjugate ofthe function q. When we apply the reduction (3) to the Eqs.(1)and(2)weobtainthecomplexmodiﬁedKorteweg–deVries(cmKdV)equation[2–4]

aqt=−1 4qxxx+3

2k¯qqqx, (5)

providedthat ¯a=a.Thesecondreduction(4)givestheusualmKdVequation[5]

aqt=−1 4qxxx+3

2kq²qx, (6)

withnoconditionona.

Most of the integrable nonlinear equations are local that is the solution’s behavior depends only on its local space andtimeparameters.In[1,7,8]AblowitzandMusslimaniintroducedintegrablenonlocalreductionswhichyield thespace- timereflectionsymmetric(ST-symmetric),thespacereflectionsymmetric(S-symmetric),andtimereflectionsymmetric(T- symmetric)equations.ForinstanceintheS-symmetriccase,thesolution’sbehavioratlocation(t,x)dependsontheinfor- mationnotonlyatthepoint(t,x)butalsoatthepoint(^t,−x)^.Âblowitzând^Musslimaniîntroduced^theST-symmetricand T-symmetricnonlocal mKdVandcmKdVequationsandthey onlyobtainedone-solitonsolutions ofST-symmetriconesby usinginversescatteringtransformin[8].Anonlocalreductionisgivenby

r

(

t,x

)

=k¯q

( ε

¹^t,

ε

²^x

)

, (7)

where

ε

²1=

ε

²2=1.UnderthisconditionthemKdVsystem(1)and(2)reduceto aqt

(

^t,x

)

=−1

4qxxx

(

^t,x

)

+3

2k¯q

( ε

¹^t^,

ε

²^x

)

^q

(

^t,x

)

^qx

(

^t,x

)

, (8) provided that ¯a=

ε

1

ε

2a.The casefor(

ε

1,

ε

2)=(¹,1) ^yields ^the^local^equation ⁽⁵⁾^.^There ^are ^three^different^nonlocal ^reductions where (

ε

1,

ε

2)=

{

(−1,1),(¹,−1),(−1,−1)

}

^. ^Hence ^for ^these ^values ^of

ε

1 and

ε

2 and for different signs of k (sign(k)=± 1), we have sixdifferent nonlocal integrable cmKdVequations obtainedby Ablowitz–Musslimani type reduction(7)whicharerespectivelyT-symmetric,S-symmetric,andST-symmetricnonlocalcmKdVequationsgivenbelowinpart A.

A. r(^t,x)=k¯q(

ε

1t,

ε

2x)^(Nonlocal^cmKdV^equations) 1.T-symmetriccmKdVequation:

aqt

(

^t,^x

)

⁼⁻¹₄^q^xxx

(

^t^,^x

)

⁺³₂^k^¯q

(

^−t^,^x

)

^q

(

^t^,^x

)

^q^x

(

^t^,^x

)

^, ^¯a⁼^−a. ⁽⁹⁾

2.S-symmetriccmKdVequation:

aqt

(

t,x

)

=−1

4qxxx

(

^t,x

)

+3

2k¯q

(

^t,−x

)

^q

(

^t,x

)

^qx

(

^t,x

)

, ¯a=−a. (10) 3.ST-symmetriccmKdVequation:

aqt

(

^t,^x

)

⁼⁻¹₄^q^xxx

(

^t^,^x

)

⁺³₂^k^¯q

(

^−t^,^−x

)

^q

(

^t^,^x

)

^q^x

(

^t^,^x

)

^, ^¯a⁼^a. ⁽¹¹⁾

ThesecondnonlocalreductionofthemKdVsystemisgivenby

r

(

^t,x

)

=kq

( ε

¹^t,

ε

²^x

)

, (12)

yieldingtheequation aqt

(

^t,x

)

=−1

4qxxx

(

^t,x

)

+3

2kq

(

^t,x

)

^q

( ε

1t,

ε

2x

)

^q^x

(

^t,x

)

, (13) providedthat

ε

1

ε

2=1.Thereforewehaveonlyonepossibility(

ε

1,

ε

2)=(−1,−1)^to^have^a^nonlocal^equation,^without anyadditionalconditiononthe parametera.TheST-symmetricnonlocal mKdVequationobtainedhereisgivenbelow inpartB.

B. r(^t,x)=kq(

ε

1t,

ε

2x)^(Nonlocal^mKdV^equation) 1.ST-symmetricmKdVequation:

aqt

(

t,x

)

=−1

4qxxx

(

^t,x

)

+3

2kq

(

−t,−x

)

^q

(

^t,x

)

^qx

(

^t,x

)

, (14) withnoconditionona.

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ThenonlocalcmKdVandnonlocalmKdVequationshavethefocusinganddefocusingcaseswhenk<0andk>0,respec- tively.Alltheaboveequationsareintegrable.

Thereisanincreasinginterestinobtainingthenonlocalreductionsofsystemsofintegrableequationsandanalyzingtheir solutionsandproperties[9–24]afterAblowitzandMusslimani’sworks[1]and[7,8]inwhichtheyproposedmanynonlocal nonlinearintegrableequationssuch asnonlocalnonlinear Schrödinger(NLS) equation,cmKdVandmKdVequations,sine- Gordonequation, (¹+1) ^and(²+1)dimensionalthree-wave interaction, Davey–Stewartsonequation,andderivative NLS equation.TheydiscussedLaxpairs,conservationlaws,inversescatteringtransforms,andobtainedone-solitonsolutions of someoftheseequations.InRef.[25],Maetal.showedthat ST-symmetricnonlocalcmKdVequationisgaugeequivalentto aspin-like modelwhichshowsthat thereexists signiﬁcantdifference betweenthenonlocal cmKdVandthelocalcmKdV equation.TheyconstructedDarbouxtransformationsfornonlocalcmKdVandobtaineddifferenttypeofexactsolutionsin- cluding dark-soliton, W-type soliton, M-typesoliton, andperiodic solutions. Ji andZhu obtained soliton, kink, anti-kink, complexiton,breather,rogue-wavesolutions,andnonlocalizedsolutionswithsingularitiesofST-symmetricnonlocalmKdV equationthroughDarbouxtransformationandinversescatteringtransform[26,27].In[28],theauthorsshowedthatmany nonlocalintegrableequations,suchasDavey–Stewartsonequation,T-symmetricNLSequation,nonlocalderivativeNLSequa- tion,andST-symmetriccmKdVequationcanbeconvertedtolocalintegrableequationsbysimplevariabletransformations.

Theyusedthesetransformationstoobtainsolutionsofthenonlocalequationsfromthesolutionsofthelocalequationsand toderive new nonlocalintegrable equations,such ascomplex andrealST- andT-symmetric NLS equationsand nonlocal complexshortpulseequations.Forsome possibleapplicationofnonlocalNLSandnonlocalmKdVequationsonecancheck [29–31].Someofoursolutionscoincidewiththesolutionsgivenin[8,25,27,28].

The main purpose of thiswork isto search for possible integrablenonlocal reductions of the mKdV system(1) and (2) andfind their soliton solutions by the application of the Hirota direct method. To find the soliton solutions of the nonlocal mKdVand nonlocal cmKdVequations we first findsoliton solutions ofthe mKdV system(1) and(2).By using thereduction formulas (7) and(12) we find thesoliton solutions ofthe nonlocal mKdV andnonlocal cmKdVequations.

Actually,herewe introduce a generalmethod forﬁnding solitonsolutions ofnonlocal integrableequations.If a nonlocal equationisconsistentlyobtainedbyanonlocalreductionofasystemofequationshavingsolitonsolutionseitherbyHirota directmethod orby other techniques,such asthe inversescattering transformtechnique, one can automaticallyuse the reductionformulas(constraintequations)toﬁndthesolitonsolutionsofthereducednonlocalequations.

Inaprevious paper[32],wehavestudied thesolitonsolutionsofthe NLSsystemandnonlocalNLS equations.Inthis work,westudysolitonsolutionsofthenonlocalmKdVequationsofalltypes.FollowingtheworkofIwaoandHirota[2]we ﬁrstﬁndone-,two-,andthree-solitonsolutionsofthecoupledmKdVsystem(1)and(2)byusingtheHirotadirectmethod.

Then byusing theAblowitz–Musslimani type reductions(7)and(12),we obtainsolitonsolutions ofthe nonlocalcmKdV (includingT-, S-,and ST-symmetricequations) andnonlocal ST-symmetricmKdV equations.We show that there aretwo different types ofone-soliton solution of the reduced mKdV system. We give the corresponding two- and three-soliton solutionsoftheﬁrsttype.Wealsopresentthegraphsofsome solutionsforcertainvaluesoftheparameters.Theyinclude one-,two-,andthree-solitonwaves,complexitons,breather-type,andkink-typewaves.

Thelayoutofthepaperisasfollows.InSection2weapplyHirotamethodtothecoupledmKdVsystem(1)and(2)and ﬁndsolitonsolutions.In Section3weﬁndsolitonsolutionsofT-symmetric,S-symmetric,andtwodifferentST-symmetric mKdVequationsandwegivesomeexamplesforone-,two-,andthree-solitonsolutionstogetherwiththeirgraphs.

2. HirotamethodforthecoupledMKdVsystem

FollowingtheworkofIwaoandHirota[2]letq= F

f andr=G

f.Eq.(1)becomes 4aFtf³− 4aFftf²+Fxxxf³− 3Fxxfxf²+6Fxf_x²f− 3Fxfxxf²

− 6Ff_x³+6Ffxfxxf− Ffxxxf²− 6GFFxf+6GF²fx=0, (15) whichisequivalentto

f²

(

⁴^aDt+D³_x

)

^F· f− 3

(

^D²xf· f+2GF

)(

^DxF· f

)

=0. Similarly,theEq.(2)canbewrittenas

f²

(

⁴^aDt+D³_x

)

^G· f− 3

(

^D²xf· f+2GF

)(

^DxG· f

)

=0. HencetheHirotabilinearformofthemKdVsystemis

P1

(

^D

) {

^F^{· f}

}

^≡

(

⁴^aDt+D³_x− 3

α

^D^x

) {

^F^{· f}

}

⁼⁰ ⁽¹⁶⁾

P2

(

^D

) {

^G^{· f}

}

^≡

(

⁴^aDt+D³_x− 3

α

^D^x

) {

^G^{· f}

}

⁼⁰ ⁽¹⁷⁾

P3

(

^D

) {

^f^{· f}

}

^≡

(

^D²x−

α ) {

^f^{· f}

}

⁼⁻²^GF^, ⁽¹⁸⁾

where

α

îsân ârbitrary^constant. ^Note^that^for^mKdV^system^we ôbtain^similar ^solutionsâsⁱⁿ^the^NLS ^system^case^[32]^.

ForadetailedworkoftheapplicationoftheHirotamethodtothemKdVsystemonecancheck[33].

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2.1. One-solitonsolutionoftheMKdVsystem

Toﬁndone-solitonsolutionweusethefollowingexpansionsforthefunctionsF,G,andf,

F=

ε

^F1, G=

ε

^G1, f=1+

ε

²^f2, (19)

where

F1=e^θ¹, G1=e^θ²,

θ

i=k_ix+

ω

it+

δ

i,i=1,2. (20)

Weinserttheseexpansionsinto(16)–(18).Thecoeﬃcientof

ε

⁰^gives

(

^D²x−

α ) {

¹^{· 1}

}

⁼⁰ ⁽²¹⁾

yieldingthat

α

=0.Analyzingthecoeﬃcientsof

ε

ⁿ^,¹≤ n≤ 4givethedispersionrelations

ω

ⁱ⁼⁻₄^k³ⁱ_a^, ⁱ⁼¹^,²^, ⁽²²⁾

andthefunctionf₂

f2=−e⁽^k¹⁺^k²⁾^x⁺^(ω¹⁺^ω²⁾^t⁺^δ¹⁺^δ²

(

^k1+k2

)

² ^. ⁽²³⁾

Take

ε

=1.HenceapairofsolutionsofthemKdVsystem(1)and(2)isgivenby(q(t,x),r(t,x))where

q

(

^t^,^x

)

⁼₁₊^e_Ae^θ¹_θ₁₊_θ₂^, ^r

(

^t,^x

)

⁼₁₊^e_Ae^θ²_θ₁₊_θ₂^, ⁽²⁴⁾

with

θ

i=k_ix−k³_i

4at+

δ

i,i=1,2,andA=− 1

(^k1+k2)²^.^Here^k¹^,^k²^,

δ

1,and

δ

2 arearbitrarycomplexnumbers.

2.2. Two-solitonsolutionoftheMKdVsystem

Fortwo-solitonsolution,wetake

F=

ε

^F¹⁺

ε

³^F³^, ^G⁼

ε

^G¹⁺

ε

³^G³^, ^f ⁼¹⁺

ε

²^f²⁺

ε

⁴^f⁴^, ⁽²⁵⁾

where

F1=e^θ¹+e^θ², G1=e^η¹+e^η², (26)

with

θ

i=k_ix+

ω

it+

δ

i,

η

i=ix+m_it+

α

i fori=1,2. When we insert above expansions into(16)–(18) andconsider the coeﬃcientsof

ε

ⁿ^,¹≤ n≤ 8weobtainthedispersionrelations

ω

i=−k³_i

4a, m_i=−³_i

4a, i=1,2, (27)

thefunctionf₂,

f2=e^θ¹⁺^η¹⁺^α¹¹+e^θ¹⁺^η²⁺^α¹²+e^θ²⁺^η¹⁺^α²¹+e^θ²⁺^η²⁺^α²² =

1≤i, j≤2

e^θⁱ⁺^η^j⁺^α^{i j}, (28)

where

e^α^{i j}=− 1

(

^ki+j

)

²^,¹^{≤ i,}^j^{≤ 2}^, ⁽²⁹⁾

thefunctionsF₃andG₃,

F3=

γ

¹^e^θ¹⁺^θ²⁺^η¹⁺

γ

²^e^θ¹⁺^θ²⁺^η²^, ^G³⁼

β

¹^e^θ¹⁺^η¹⁺^η²⁺

β

²^e^θ²⁺^η¹⁺^η²^, ⁽³⁰⁾

where

γ

i=−

(

^k1− k2

)

²

(

^k1+i

)

²

(

^k2+i

)

²^,

β

i=−

(

1− 2

)

²

(

1+ki

)

²

(

2+ki

)

²^, ⁱ⁼¹^,²^, ⁽³¹⁾

andthefunctionf4

f4=Me^θ¹⁺^θ²⁺^η¹⁺^η², (32)

where

M=

(

^k1− k2

)

²

(

^l1− l2

)

²

(

^k1+l1

)

²

(

^k1+l2

)

²

(

^k2+l1

)

²

(

^k2+l2

)

²^. ⁽³³⁾

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Letusalsotake

ε

=1.Thentwo-solitonsolutionofthemKdVsystem(1)and(2)isgivenwiththepair(q(t,x),r(t,x)), q

(

^t,x

)

= e^θ¹+e^θ²+

γ

1e^θ¹⁺^θ²⁺^η¹+

γ

2e^θ¹⁺^θ²⁺^η²

1+e^θ¹⁺^η¹⁺^α¹¹+e^θ¹⁺^η²⁺^α¹²+e^θ²⁺^η¹⁺^α²¹+e^θ²⁺^η²⁺^α²²+Me^θ¹⁺^θ²⁺^η¹⁺^η², (34)

r

(

^t,x

)

= e^η¹+e^η²+

β

1e^θ¹⁺^η¹⁺^η²+

β

2e^θ²⁺^η¹⁺^η²

1+e^θ¹⁺^η¹⁺^α¹¹+e^θ¹⁺^η²⁺^α¹²+e^θ²⁺^η¹⁺^α²¹+e^θ²⁺^η²⁺^α²²+Me^θ¹⁺^θ²⁺^η¹⁺^η², (35) with

θ

i=k_ix−k³_i

4at+

δ

i,

η

i=ix− ³_i

4at+

α

ifori=1,2.Herek_i,i,

δ

i,and

α

i,i=1,2arearbitrarycomplexnumbers.

2.3.Three-solitonsolutionoftheMKdVsystem

Toﬁndthree-solitonsolution,wetake

f=1+

ε

²^f²⁺

ε

⁴^f⁴⁺

ε

⁶^f⁶^, ^G⁼

ε

^G¹⁺

ε

³^G³⁺

ε

⁵^G⁵^, ^F⁼

ε

^F¹⁺

ε

³^F³⁺

ε

⁵^F⁵^, ⁽³⁶⁾

and

F1=e^θ¹+e^θ²+e^θ³, G1=e^η¹+e^η²+e^η³, (37)

where

θ

i=k_ix+

ω

it+

δ

i,

η

i=ix+m_it+

α

i fori=1,2,3.Inserting(36)into(16)–(18)andanalyzingthecoeﬃcientsof

ε

ⁿ^,

1≤ n≤ 12givethedispersionrelations

ω

i=−k³_i

4a, m_i=−³_i

4a, i=1,2,3, (38)

thefunctionf₂

f2=

1≤i, j≤3

e^θⁱ⁺^η^j⁺^α^{i j}, e^α^{i j}=− 1

(

^ki+j

)

²^, ¹^{≤ i,}^j^{≤ 3}^, ⁽³⁹⁾

thefunctionsF₃ andG₃

F3=

1≤i, j,s≤3 i< j

Ai jse^θⁱ⁺^θ^j⁺^η^s, Ai js=−

(

^ki− kj

)

²

(

^ki+s

)

²

(

^kj+s

)

²^{, 1}^{≤ i,}^j,^s^{≤ 3}^, ⁱ^<^j, ⁽⁴⁰⁾

G3=

1≤i, j,s≤3 i< j

Bi jse^ηⁱ⁺^η^j⁺^θ^s, Bi js=−

(

i− j

)

²

(

i+ks

)

²

(

j+ks

)

²^{, 1}^{≤ i,}^j,^s^{≤ 3}^, ⁱ^< ^j, ⁽⁴¹⁾

thefunctionf₄ f4=

1≤i< j≤3 1≤p<r≤3

Mi jpre^θⁱ⁺^θ^j⁺^η^p⁺^η^r, (42)

where

M_{i jpr}=

(

^ki− kj

)

²

(

^lp− lr

)

²

(

^ki+lp

)

²

(

^ki+lr

)

²

(

^kj+lp

)

²

(

^kj+lr

)

²^, ⁽⁴³⁾

for1≤ i<j≤ 3,1≤ p<r≤ 3,andthefunctionsF₅andG₅

F5=V12e^θ¹⁺^θ²⁺^θ³⁺^η¹⁺^η²+V13e^θ¹⁺^θ²⁺^θ³⁺^η¹⁺^η³+V23e^θ¹⁺^θ²⁺^θ³⁺^η²⁺^η³, (44)

G5=W12e^θ¹⁺^θ²⁺^η¹⁺^η²⁺^η³+W13e^θ¹⁺^θ²⁺^η¹⁺^η²⁺^η³+W23e^θ²⁺^θ³⁺^η¹⁺^η²⁺^η³, (45) where

Vi j= S_{i j}

4a

( ω

¹⁺

ω

²⁺

ω

³⁺^mⁱ⁺^m^j

)

+

(

^k1+k2+k3+i+j

)

³^, ⁽⁴⁶⁾

W_{i j}= Q_{i j}

4a

( ω

i+

ω

j+m1+m2+m3

)

+

(

^ki+k_j+1+2+3

)

³^, ⁽⁴⁷⁾

for1≤ i<j≤ 3.HereS_ijandQ_ijaregiveninAppendixofRef.[33].Wealsoobtainthefunctionf₆as

f6=He^θ¹⁺^θ²⁺^θ³⁺^η¹⁺^η²⁺^η³, (48)

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wherethe coeﬃcient Hisalsogivenin AppendixofRef. [33]. Letusalsotake

ε

=1.Hence three-solitonsolutionof the coupledmKdVsystem(1)and(2)isgivenwiththepair(q(t,x),r(t,x))where

q

(

^t,x

)

=

e^θ¹+e^θ²+e^θ³+

1≤i, j,s≤3 i< j

A_{i js}e^θⁱ⁺^θ^j⁺^η^s+

1≤i, j≤3 i< j

V_{i j}e^θ¹⁺^θ²⁺^θ³⁺^ηⁱ⁺^η^j 1+

1≤i, j≤3e^θⁱ⁺^η^j⁺^α^{i j}+

1≤i< j≤3 1≤p<r≤3

M_{i jpr}e^θⁱ⁺^θ^j⁺^η^p⁺^η^r+He^θ¹⁺^θ²⁺^θ³⁺^η¹⁺^η²⁺^η³, (49)

r

(

^t,^x

)

⁼

e^η¹+e^η²+e^η³+

1≤i, j,s≤3 i< j

Bi jse^ηⁱ⁺^η^j⁺^θ^s+

1≤i, j≤3 i< j

Wi je^θⁱ⁺^θ^j⁺^η¹⁺^η²⁺^η³ 1+

1≤i, j≤3e^θⁱ⁺^η^j⁺^α^{i j}+

1≤i< j≤3 1≤p<r≤3

Mi jpre^θⁱ⁺^θ^j⁺^η^p⁺^η^r+He^θ¹⁺^θ²⁺^θ³⁺^η¹⁺^η²⁺^η³. (50)

Havingobtainedtheone-, two-,andthree-solitonsolutionsofthe mKdVsystemwenow readytoobtain such soliton solutionsofthenonlocalreductionsofthemKdVsystem.SolitonsolutionsofthelocalreductionsofthemKdVsystemcan befoundin[33].Hereinoursolutionswefocusonthedomaint≥ 0,x∈R.

3. NonlocalreductionsoftheMKdVsystem

Toﬁndthesolitonsolutionsofthenonlocalintegrableequationswhichareobtainedbyconsistentnonlocal reductions ofanintegrablesystemofequationsweusethefollowingthreesteps.

(i) Findconsistentreductionformulaswhichreducetheintegrablesystemequationstointegrablenonlocalequations.

(ii) FindsolitonsolutionsofthesystemequationsbyuseoftheHirotadirectmethodorby inversescatteringtransform technique,orbyuseofDarbouxTransformation.

(iii) Use thereduction formulas on thesoliton solutions ofthe systemequationsto obtain the solitonsolutions ofthe reducednonlocal equations.Bythiswayone obtains manydifferentrelationsamong thesolitonparameters ofthe systemequations.

Inthissection, usingtheabovemethod,we willﬁrstusethereduction (7)givenbyAblowitz andMusslimani [1]and [7,8]andobtainsolitonsolutionsforthreedifferentnonlocalcmKdVequations(9)–(11)withthecondition

¯a=

ε

1

ε

2a (51)

satisﬁed.Secondly,we willdealwiththe reduction(12)andobtain one-andtwo-solitonsolutions ofthenonlocalmKdV equation(14).

3.1. One-solitonsolutionforthenonlocalCMKdVequations:(^r=k¯q(

ε

1t,

ε

2x))

Wehavetwotypesofsolitonsolutionsofthereducednonlocalequations.Themainideahereistousetheone-soliton solutions (24)ofthemKdVsystemequations(1)–(2)andthen usethe reductionformulas(7)and(12).Bythisprocedure weobtaintwotypesofsolitonsolutionsofthereducednonlocalequations.

3.1.1. Type1

Firstly, we ﬁndtheconditions ontheparameters ofone-soliton solutionofthemKdV systemtosatisfy the constraint equation(7).Usingthisconstraintequationweget

e^k²^x⁻^k

3 2 4at+δ² 1+Ae⁽^k¹⁺^k²⁾^x⁻⁽^k

3 1+k3

2) 4a t+δ1+δ2

=k e^¯k¹^ε²^x⁻^¯k

3 1 4¯aε¹t+δ¯¹

1+A¯e⁽^¯k¹⁺^¯k²⁾^ε²^x⁻⁽^¯k

3 1+¯k3

2) 4¯a ε1t+δ¯1+δ¯2

. (52)

Thisequation givestwo differentrelationsamongthesolitonparameters. Oneofthecase(type-1) includesthefollowing equalitiesthatmustbesatisﬁedbytheparameters:

i

)

^k2=

ε

²^¯k¹^, ⁱⁱ

)

₄^k³²_a= ¯k³₁

4¯a

ε

¹^, ⁱⁱⁱ

)

^e^δ²=ke^δ^¯¹, i

v )

^A=A¯,

v ₎ ₍

k1+k2

)

=

(

^¯k1+¯k₂

) ε

2,

v

i

) (

^k³1+k³₂

)

4a =

(

^¯k³1+¯k³₂

)

4¯a

ε

1,

v

ii

)

^e^δ¹⁺^δ²=e^δ^¯¹⁺^δ^¯². (53) If we usethe conditions (51) andi) onthe left hand side of the equalityii), it isclear that this equalityis satisﬁed directlysince

k³₂

4a=

ε

²^¯k³1

4

ε

1

ε

2¯a= ¯k³₁ 4¯a

ε

1.

Withtheconditiongivenini)itisobviousthativ)issatisﬁeddirectlysince

− 1

(

^k1+k2

)

² ⁼⁻ 1

(

^¯k2

ε

2+¯k₁

ε

2

)

² ⁼⁻ 1

(

^¯k1+¯k₂

)

²^.

(7)

Theconditionv)issatisﬁedsince

(

^k1+k2

)

=

(

^¯k2

ε

²⁺^¯k¹

ε

²

)

=

(

^¯k1+¯k2

) ε

²

bytheconditionk₂=

ε

2¯k₁orequivalentlyk₁=

ε

2¯k₂.Bythesamemannervi)isalreadytruesince

(

^k³1+k³₂

)

4a =

(

^¯k³2+¯k³₁

)

4

ε

¹

ε

²^¯a ⁼

⁽

¯k³₁+¯k³₂

)

4¯a

ε

¹^.

Finally,considertherelatione^δ²=ke^δ^¯¹ ore^δ^¯²=ke^δ¹ giveninvii).Sincekisarealconstantwehavee^δ¹⁺^δ²=ke^δ¹e^δ^¯¹ and e^δ^¯¹⁺^δ^¯²=ke^δ^¯¹e^δ¹ thatyieldtheequalitye^δ¹⁺^δ²=e^δ^¯¹⁺^δ^¯².

Thustheparametersofone-solitonsolutionoftheEq.(8)musthavethefollowingproperties:

1

)

^¯a⁼

ε

¹

ε

²^a, ²

)

^k²⁼

ε

²^¯k¹^, ³

)

^e^δ²⁼^ke^δ^¯¹^. ⁽⁵⁴⁾

Thecase(

ε

1,

ε

2)=(¹,1)^gives^localêquation.^For^particular^choiceôf^the^parameters^let ûs^check ^the^solutionsôf^the nonlocalreductionsofthemKdVsystemfor(

ε

1,

ε

2)=

{

(−1,1),(¹,−1),(−1,−1)

}

^.

3.1.1.1. Casea.(T-symmetric). r=k¯q(−t,x)^.^This^case^gives ^¯a=−a,k₂=¯k₁,and aqt

(

t,x

)

=−1

4qxxx

(

^t,x

)

+3

2k¯q

(

−t,x

)

^q

(

^t,x

)

^qx

(

^t,x

)

, (55) withe^δ²=ke^δ^¯¹.Since ¯a=−a,aispureimaginarysaya=ib,fornonzerob∈R.Letk₁=

α

+i

β

^so^k2=

α

− i

β

^for

α

,

β

∈R,

α

=0.Thenthesolutionof(55)becomes q

(

^t,^x

)

⁼ ^e^(α⁺ⁱ^β)^x⁺

(β³−3α²β)+i(α³−3αβ²)

4b t+δ¹

1−₄^k_α2e²^α^x⁺ⁱ^α³⁻³²^b^αβ²^t⁺^δ¹⁺^δ^¯¹

. (56)

Thissolutionisalsogivenin[28].Thecorrespondingfunction|q(t,x)|²is

|

^q

(

^t,x

) |

²= e²^α^x⁺^(β

3−3α²β) 2b t+δ1+δ¯1

k

4α²e²^α^x⁺^δ¹⁺^δ^¯¹− cos

(

^(α³⁻³2b^αβ²⁾t

)

2

+sin²

(

^(α³⁻³2b^αβ²⁾t

)

. (57)

When

α

³− 3

αβ

²=0andt= _α3²−3^nbαβ^π ², ₄_α^k2e²^α^x⁺^δ¹⁺^δ^¯¹−(−1)ⁿ=0wherenisaninteger,forbothfocusinganddefocusing cases,thesolutionissingular.When

α

³− 3

αβ

²=0thesolutionforfocusingcaseisnon-singular.When

α

=0thesolution isexponentially growing for ^β_b³ >0 andexponentially decayingfor ^β_b³ <0.Now forparticular choicesoftheparameters satisfyingtheconditions(54)wegiveanexampleofasolutionoftheEq.(55)andpresentthegraphofthesolution.

Example1. Fortheparameters(^k1,k₂,e^δ¹,e^δ²,k,a)=(²^√³+2i,2√

3− 2i,1+i,−1+i,−1,10i)^we^obtain^thenon-singular solutionof(55)as

q

(

t,x

)

= 24

(

¹+i

)

^e⁽²^√³⁺²ⁱ⁾^x⁻⁸⁵^t

24+e⁴^√³^x , (58)

sothefunction|q(t,x)|² is

|

^q

(

^t,x

) |

²=12e⁻¹⁶⁵^tsech²

(

²^√³^x+

δ )

, (59)

where

δ

=−¹₂ln(²⁴)^.^The^solution^is^anasymptoticallydecayingsolutionfort>0.Thegraphof(59)isgiveninFig.1.

3.1.1.2.Caseb.(S-symmetric):r=k¯q(^t,−x)^. ^In^this^case^we^have ^¯a=−a,k₂=−¯k1,and

aqt

(

^t,^x

)

⁼⁻¹₄^q^xxx

(

^t^,^x

)

⁺³₂^k^¯q

(

^t^,^−x

)

^q

(

^t^,^x

)

^q^x

(

^t^,^x

)

⁽⁶⁰⁾

withe^δ²=ke^δ^¯¹.Since ¯a=−a,itispure imaginary,say a=ib for nonzerob∈R.Letalsok₁=

α

+i

β

^and^so ^k2=−

α

+i

β

for

α

,

β

∈R,

β

=0.Thenthesolutionof(60)becomes q

(

^t,^x

)

⁼ ^e^(α⁺ⁱ^β)^x⁺

(β³−3α²β)+i(α³−3αβ²)

4b t+δ¹

1+₄_β^k2e²ⁱ^β^x⁺ⁱ^α³⁻³²^b^αβ²^t⁺^δ¹⁺^δ^¯¹

, (61)

andsothefunction|q(t,x)|² is

|

^q

(

^t,x

) |

²= e²^α^x⁺^(β

3−3α²β) 2b t+δ1+δ¯1

k

4β²e^(β³⁻³²^b^α²^β)^t+^δ¹⁺^δ^¯¹+cos

(

²

β

^x

)

2

+sin²

(

²

β

^x

)

. (62)