research article time-dependent toroidal compactification ...research article time-dependent...

Research ArticleTime-Dependent Toroidal Compactification Proposals and theBianchi Type I Model Classical and Quantum Solutions

L Toledo Sesma J Socorro and O Loaiza

Departamento de Fısica DCI Universidad de Guanajuato Campus Leon Loma del Bosque No 103 Colonia Lomas del CampestreApartado Postal E-143 37150 Leon GTO Mexico

Correspondence should be addressed to J Socorro socorrofisicaugtomx

Received 19 January 2016 Revised 22 April 2016 Accepted 5 May 2016

Academic Editor Elias C Vagenas

Copyright copy 2016 L Toledo Sesma et alThis is an open access article distributed under theCreativeCommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited Thepublication of this article was funded by SCOAP3

We construct an effective four-dimensional model by compactifying a ten-dimensional theory of gravity coupled with a real scalardilaton field on a time-dependent torus This approach is applied to anisotropic cosmological Bianchi type I model for whichwe study the classical coupling of the anisotropic scale factors with the two real scalar moduli produced by the compactificationprocess Under this approach we present an isotropization mechanism for the Bianchi I cosmological model through the analysisof the ratio between the anisotropic parameters and the volume of the Universe which in general keeps constant or runs into zerofor late times We also find that the presence of extra dimensions in this model can accelerate the isotropization process dependingon the momenta moduli values Finally we present some solutions to the correspondingWheeler-DeWitt (WDW) equation in thecontext of standard quantum cosmology

1 Introduction

The 2015 release of Planck data has provided a detailedmap of cosmic microwave background (CMB) temperatureand polarization allowing us to detect deviations from anisotropic early Universe [1] The evidence given by these dataleads us to the possibility of considering that there is no exactisotropy since there exist small anisotropy deviations of theCMB radiation and apparent large angle anomalies In thatcontext there have been recent attempts to fix constraintson such deviations by using the Bianchi anisotropic models[2] The basic idea behind these models is to consider thepresent observational anisotropies and anomalies as imprintsof an early anisotropic phase on the CMB which in turncan be explained by the use of different Bianchi models Inparticular Bianchi I model seems to be related to large angleanomalies [2] (and references therein)

In a different context attempts to understand diverseaspects of cosmology as the presence of stable vacua andinflationary conditions in the framework of supergravity andstring theory have been considered in the last years [3ndash10]One of the most interesting features emerging from these

types of models consists in the study of the consequencesof higher dimensional degrees of freedom on the cosmologyderived from four-dimensional effective theories [11 12]The usual procedure for that is to consider compactificationon generalized manifolds on which internal fluxes haveback-reacted altering the smooth Calabi-Yau geometry andstabilizing all moduli [10]

The goal of the present work is to consider in a simplemodel some of the above two perspectives that is we willconsider the presence of extra dimensions in a Bianchi Imodel with the purpose of tracking down the influence ofmoduli fields in its isotropization For that wewill consider analternative procedure concerning the role played by moduliIn particular we will not consider the presence of fluxesas in string theory in order to obtain a moduli-dependentscalar potential in the effective theory Rather we are goingto promote some of the moduli to time-dependent fields byconsidering the particular case of a ten-dimensional gravitycoupled to a time-dependent dilaton compactified on a 6-dimensional torus with a time-dependent Kahler modulusWith the purpose to study the influence of such fields weare going to ignore the dynamics of the complex structure

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2016 Article ID 6705021 12 pageshttpdxdoiorg10115520166705021

2 Advances in High Energy Physics

modulos (for instance by assuming that it is already stabilizedby the presence of a string field in higher scales) This willallow us to construct classical effective models with twomoduli

However we are also interested in possible quantumaspects of our model Quantum implications on cosmologyfrommore fundamental theories are expected due to differentobservations For instance it has been pointed out thatthe presence of extra dimensions leads to an interestingconnection to the ekpyrotic model [13] which generatedconsiderable activity [8 9 14] The essential ingredient inthesemodels (see for instance [11]) is to consider an effectiveaction with a graviton and a massless scalar field the dilatondescribing the evolution of the Universe while incorporatingsome of the ideas of pre-big-bang proposal [15 16] in thatthe evolution of the Universe began in the far past Onthe other hand it is well known that relativistic theoriesof gravity such as general relativity or string theories areinvariant under reparametrization of time Quantization ofsuch theories presents a number of problems of principleknown as ldquothe problem of timerdquo [17 18] This problem ispresent in all systems whose classical version is invariantunder time reparametrization leading to its absence at thequantum level Therefore the formal question involves howto handle the classical Hamiltonian constraintH asymp 0 in thequantum theory Also connected with the problem of time isthe ldquoHilbert space problemrdquo [17 18] referring to the not at allobvious selection of the inner product of states in quantumgravity and whether there is a need for such a structure at allThe above features as it is well known point out the necessityto construct a consistent theory of gravity at quantum level

Analyses of effective four-dimensional cosmologies de-rived from M-theory and Type IIA string theory wereconsidered in [19ndash21] where string fluxes are related to thedynamical behavior of the solutions A quantum descriptionof the model was studied in [22 23] where a flat Friedmann-Robertson-Walker geometry was considered for the extendedfour-dimensional space-time while a geometry given by 1198781 times119879

6 was assumed for the internal seven-dimensional spacehowever the dynamical fields are in the quintom schemesince one of the fields has a negative energy Under a similarperspective we study the Hilbert space of quantum states ona Bianchi I geometry with two time-dependent scalar moduliderived from a ten-dimensional effective action containingthe dilaton and the Kahler parameter from a six-dimensionaltorus compactification We find that in this case the wavefunction of this Universe is represented by two factors onedepending onmoduli and the second one depending on grav-itational fieldsThis behavior is a property of all cosmologicalBianchi Class A models

The work is organized in the following form In Section 2we present the construction of our effective action by com-pactification on a time-dependent torus while in Section 3we study its Lagrangian and Hamiltonian descriptions usingas toymodel the Bianchi type I cosmologicalmodel Section 4is devoted to finding the corresponding classical solutionsfor few different cases involving the presence or absence ofmatter content and the cosmological constant term Using

the classical solutions found in previous sections we presentan isotropization mechanism for the Bianchi I cosmolog-ical model through the analysis of the ratio between theanisotropic parameters and the volume of the Universeshowing that in all the cases we have studied its valuekeeps constant or runs into zero for late times In Section 5we present some solutions to the corresponding Wheeler-DeWitt (WDW) equation in the context of standard quantumcosmology and finally our conclusions are presented inSection 6

2 Effective Model

We start from a ten-dimensional action coupled with adilaton (which is the bosonic component common to allsuperstring theories) which after dimensional reduction canbe interpreted as a Brans-Dicke like theory [24] In thestring frame the effective action depends on two spacetime-dependent scalar fields the dilaton Φ(119909120583) and the Kahlermodulus 120590(119909120583) For simplicity in this work we will assumethat these fields only depend on time The high-dimensional(effective) theory is therefore given by

119878 =

1

21205812

10

int119889

10119883radicminus119866119890

minus2Φ[R

(10)

+ 4119866

119872119873

nabla119872Φnabla

119873Φ]

+ int119889

10119883radicminus119866L

119898

(1)

where all quantities refer to the string frame while the ten-dimensional metric is described by

119889119904

2=119866119872119873119889119883

119872119889119883

119873=

120583]119889119909120583119889119909

]+ ℎ

119898119899119889119910

119898119889119910

119899 (2)

where119872119873 are the indices of the ten-dimensional spaceand Greek indices 120583 ] = 0 3 and Latin indices119898 119899 = 4 9 correspond to the external and internalspaces respectively We will assume that the six-dimensionalinternal space has the form of a torus with a metric given by

ℎ119898119899= 119890

minus2120590(119905)120575119898119899 (3)

with 120590 being a real parameterDimensionally reducing the first term in (1) to four-

dimensions in the Einstein frame (see the Appendix fordetails) gives

1198784=

1

21205812

4

int119889

4119909radicminus119892 (R minus 2119892

120583]nabla120583120601nabla]120601

minus 96119892

120583]nabla120583120590nabla]120590 minus 36119892

120583]nabla120583120601nabla]120590)

(4)

where 120601 = Φ + (12) ln(119881) with 119881 given by

119881 = 119890

6120590(119905)Vol (1198836) = int119889

6119910 (5)

By considering only a time-dependence on the moduli onecan notice that for the internal volume Vol(119883

6) to be small

the modulus 120590(119905) should be a monotonic increasing function

Advances in High Energy Physics 3

on time (recall that 120590 is a real parameter) while 119881 is time andmoduli independent

Now concerning the second term in (1) we will requireproperly defining the ten-dimensional stress-energy tensor119879119872119873

In the string frame it takes the form

119879119872119873

= (

119879120583] 0

0119879119898119899

) (6)

where 119879120583] and 119879

119898119899denote the four- and six-dimensional

components of 119879119872119873

Observe that we are not consideringmixing components of 119879

119872119873among internal and external

components although nonconstrained expressions for 119879119872119873

have been considered in [25] In the Einstein frame the four-dimensional components are given by

119879120583] = 119890

2120601119879120583] (7)

It is important to remark that there are some dilemmas aboutwhat is the best frame to describe the gravitational theoryHere in this work we have taken the Einstein frame Usefulreferences to find a discussion about string and Einsteinframes and its relationship in the cosmological context are[26ndash30]

Now with expression (4) we proceed to build theLagrangian and the Hamiltonian of the theory at the classicalregime employing the anisotropic cosmological Bianchi typeI model The moduli fields will satisfy the Klein-Gordon likeequation in the Einstein frame as an effective theory

3 Classical Hamiltonian

In order to construct the classical Hamiltonian we are goingto assume that the background of the extended space isdescribed by a cosmological Bianchi type I model For thatlet us recall here the canonical formulation in the ADMformalism of the diagonal Bianchi Class A cosmologicalmodels the metric has the form

119889119904

2= minus119873 (119905) 119889119905

2+ 119890

2Ω(119905)(119890

2120573(119905))

119894119895120596

119894120596

119895 (8)

where 119873(119905) is the lapse function 120573119894119895(119905) is a 3 times 3 diagonal

matrix 120573119894119895= diag(120573

++radic3120573

minus 120573

+minusradic3120573

minus minus2120573

+)Ω(119905) and 120573

plusmn

are scalar functions known as Misner variables and 120596119894 areone-forms that characterize each cosmological Bianchi typemodel [31] and obey the form 119889120596

119894= (12)119862

119894

119895119896120596

119895and 120596

119896 with119862

119894

119895119896the structure constants of the corresponding model The

one-forms for the Bianchi type I model are1205961 = 1198891199091205962 = 119889119910and1205963 = 119889119911 So the correspondingmetric of the Bianchi typeI in Misnerrsquos parametrization has the form

119889119904

2

119868= minus119873

2119889119905

2+ 119877

2

1119889119909

2+ 119877

2

2119889119910

2+ 119877

2

3119889119911

2 (9)

where 119877119894

3

119894=1are the anisotropic radii and they are given by

1198771= 119890

Ω+120573++radic3120573

minus

1198772= 119890

Ω+120573+minusradic3120573

minus

1198773= 119890

Ωminus2120573+

119881 = 119877111987721198773= 119890

3Ω

(10)

where 119881 is the volume function of this modelThe Lagrangian density with a matter content given by

a barotropic perfect fluid and a cosmological term has astructure corresponding to an energy-momentum tensor ofperfect fluid [32ndash36]

119879120583] = (119901 + 120588) 119906120583119906] + 119901119892120583] (11)

that satisfies the conservation law nabla]119879120583]= 0 Taking the

equation of state 119901 = 120574120588 between the energy density andthe pressure of the comovil fluid a solution is given by 120588 =119862120574119890

minus3(1+120574)Ω with 119862120574the corresponding constant for different

values of 120574 related to the Universe evolution stage Then theLagrangian density reads

Lmatt = 16120587119866119873radicminus119892120588 + 2radicminus119892Λ

= 16119873120587119866119873119862120574119890

minus3(1+120574)Ω+ 2119873Λ119890

3Ω

(12)

while the Lagrangian that describes the fields dynamics isgiven by

L119868=

119890

3Ω

119873

[6Ω

2

minus 6120573

2

+minus 6

120573

2

minus+ 96

2+ 36

120601 + 2

120601

2

+ 16120587119866119873

2120588 + 2Λ119873

2]

(13)

using the standard definition of the momenta Π119902120583 =

120597L120597120583 where 119902120583 = (Ω 120573+ 120573

minus 120601 120590) and we obtain

ΠΩ=

12119890

3ΩΩ

119873

Ω =

119873119890

minus3ΩΠΩ

12

Πplusmn= minus

12119890

3Ω120573plusmn

119873

120573plusmn= minus

119873119890

minus3ΩΠ120573plusmn

12

Π120601=

119890

3Ω

119873

[36 + 4120601]

120601 =

119873119890

minus3Ω

44

[3Π120590minus 16Π

120601]

Π120590=

119890

3Ω

119873

[192 + 36120601] =

119873119890

minus3Ω

132

[9Π120601minus Π

120590]

(14)

and introducing them into the Lagrangian density we obtainthe canonical Lagrangian asLcanonical = Π119902120583

120583minus119873H When

we perform the variation of this canonical lagrangian withrespect to 119873 120575Lcanonical120575119873 = 0 implying the constraintH

119868= 0 In our model the only constraint corresponds to


Hamiltonian density which is weakly zero So we obtain theHamiltonian density for this model

H119868=

119890

minus3Ω

24

[Π

2

Ωminus Π

2

+minus Π

2

minusminus

48

11

Π

2

120601+

18

11

Π120601Π120590

minus

1

11

Π

2

120590minus 384120587119866

119873120588120574119890

3(1minus120574)Ωminus 48Λ119890

6Ω]

(15)

We introduce a new set of variables in the gravitational partgiven by 1198901205731+1205732+1205733 = 119890

3Ω= 119881 which corresponds to the

volume of the Bianchi type I Universe in similar way to theflat Friedmann-Robetson-Walker metric (FRW) with a scalefactor This new set of variables depends on Misner variablesas

1205731= Ω + 120573

++radic3120573

minus

1205732= Ω + 120573

+minusradic3120573

minus

1205733= Ω minus 2120573

+

(16)

from which the Lagrangian density (13) can be transformedas

L119868=

119890

1205731+1205732+1205733

119873

(2120601

2

+ 36120601 + 96

2+ 2

1205731

1205732

+ 21205731

1205733+ 2

1205733

1205732+ 2Λ119873

2

+ 16120587119866119873

2120588120574119890

minus(1+120574)(1205731+1205732+1205733))

(17)

and the Hamiltonian density reads

H119868=

1

8

119890

minus(1205731+1205732+1205733)[minusΠ

2

1minus Π

2

2minus Π

2

3+ 2Π

1Π2

+ 2Π1Π3+ 2Π

2Π3minus

16

11

Π

2

120601+

6

11

Π120601Π120590minus

1

33

Π

2

120590

minus 128120587119866120588120574119890

(1minus120574)(1205731+1205732+1205733)minus 16Λ119890

2(1205731+1205732+1205733)]

(18)

So far we have built the classical Hamiltonian density froma higher dimensional theory this Hamiltonian contains abarotropic perfect fluid that we have added explicitly Thenext step is to analyze three different cases involving termsin the classical Hamiltonian (18) and find solutions to each ofthem in the classical regime

31 Isotropization The current observations of the cosmicbackground radiation set a very stringent limit to theanisotropy of the Universe [37] therefore it is importantto consider the anisotropy of the solutions We will denotethrough all our study derivatives of functions 119865 with respectto 120591 = 119873119905 by 1198651015840 (the following analysis is similar to thispresented recently in the K-essence theory since the corre-sponding action is similar to this approach [38]) Recallingthe Friedmann like equation to the Hamiltonian density

Ω

10158402= 120573

10158402

++ 120573

10158402

minus+ 16120590

10158402+

1

3

120601

10158402+ 6120601

1015840120590

1015840

+

8

3

120587119866120583120574119890

minus3(1+120574)Ω+

Λ

3

(19)

we can see that isotropization is achieved when the termswith 12057310158402

plusmngo to zero or are negligible with respect to the other

terms in the differential equationWe find in the literature thecriteria for isotropization among others (12057310158402

++120573

10158402

minus)119867

2rarr 0

(120573

10158402

++120573

10158402

minus)120588 rarr 0 that are consistent with our above remark

In the present case the comparison with the density shouldinclude the contribution of the scalar field We define ananisotropic density 120588

119886that is proportional to the shear scalar

120588119886= 120573

10158402

++ 120573

10158402

minus (20)

and will compare it with 120588120574 120588

120601 and Ω10158402 From the Hamilton

analysis we now see thatΠ120601= 0 997888rarr Π

120601= 119901

120601= constant

Π120590= 0 997888rarr Π

120590= 119901

120590= constant

Π+= 0 997888rarr Π

+= 119901

+= constant

Πminus= 0 997888rarr Π

minus= 119901

minus= constant

(21)

and the kinetic energy for the scalars fields 120601 and 120590 areproportional to 119890minus6Ω This can be seen from definitions ofmomenta associated with Lagrangian (13) and the aboveequationsThen defining 120581

Ωas 1205812

Ωsim 119901

2

++119901

2

minus+(16132

2)(119901

120590minus

9119901120601)

2+(1(3sdot44

2))(3119901

120590minus16119901

120601)

2+(1(22sdot44))(119901

120590minus9119901

120601)(3119901

120590minus

16119901120601) we have that

120588119886sim 119890

minus6Ω

120588120601120590sim 119890

minus6Ω

Ω

10158402sim

Λ

3

+ 120581Ω

2119890

minus6Ω+ 119887

120574119890

minus3(1+120574)Ω

119887120574=

8

3

120587119866119873120583120574

(22)

With the use of these parameters we find the following ratios120588119886

120588120601120590

sim constant

120588119886

120588120574

sim 119890

3Ω(120574minus1)

120588119886

Ω10158402sim

1

120581Ω

2+ (Λ3) 119890

6Ω+ 119887

1205741198903(1minus120574)Ω

(23)

We observe that for an expanding Universe the anisotropicdensity is dominated by the fluid density (with the exceptionof the stiff fluid) or by theΩ10158402 term and then at late times theisotropization is obtained since the above ratios tend to zero

4 Case of Interest

In this section we present the classical solutions for theHamiltonian density of Lagrangian (13) we have previouslybuilt in terms of a new set of variables (16) focusing on threedifferent cases We start our analysis on the vacuum caseand on the case with a cosmological term Λ Finally we willanalyze the general case considering matter content and acosmological term


41 VacuumCase To analyze the vacuumcase wewill take intheHamiltonian density (18) that 120588

120574= 0 andΛ = 0 obtaining

that

H119868vac=

1

8

119890

minus(1205731+1205732+1205733)[minusΠ

2

1minus Π

2

2minus Π

2

3+ 2Π

1Π2

+ 2Π1Π3+ 2Π

2Π3minus

16

11

Π

2

120601+

6

11

Π120601Π120590minus

1

33

Π

2

120590]

(24)

The Hamilton equations for the coordinates fields 119894=

120597H120597119875119894and the corresponding momenta

119875119894= 120597H120597119902

119894

become

Π119894= minusH equiv 0 997904rArr Π

119894= constant

Π120601= 0 997904rArr Π

120601= constant

Π120590= 0 997904rArr Π

120590= constant

120573

1015840

1=

1

4

119890

minus(1205731+1205732+1205733)[minusΠ

1+ Π

2+ Π

3]

120573

1015840

2=

1

4

119890

minus(1205731+1205732+1205733)[minusΠ

2+ Π

1+ Π

3]

120573

1015840

3=

1

4

119890

minus(1205731+1205732+1205733)[minusΠ

3+ Π

1+ Π

2]

120601

1015840= 120601

0119890

minus(1205731+1205732+1205733)

120590

1015840= 120590

0119890

minus(1205731+1205732+1205733)

(25)

The gravitational momenta are constant by mean of theHamiltonian constraint first line in the last equation In thisway the solution for the sum of 120573

119894functions becomes

1205731+ 120573

2+ 120573

3= ln [120598120591 + 119887

0] 120598 = 119887

1+ 119887

2+ 119887

3 (26)

with 119887119894= Π

119894and 119887

0being an integration constant Therefore

Misner variables are expressed as

Ω = ln (120598120591 + 1198870)

13

120573+= ln (120598120591 + 119887

0)

(120598minus31198873)6120598

120573minus= ln (120598120591 + 119887

0)

(1198873+21198871minus120598)2radic3120598

(27)

while moduli fields are given by

120601 =

1206010

120598

ln (120598120591 + 1198870)

120590 =

1205900

120598

ln (120598120591 + 1198870)

(28)

where 1206010= Π

120601and 120590

0= Π

120590 Notice that in this case

the associated external volume given by 1198903Ω and the modulifields 120601 and 120590 are all logarithmically increasing functions ontime implying for the latter that the internal volume shrinksin size for late times as expected while the four-dimensionalUniverse expands into an isotropic flat spacetimeTheparam-eter 120588

119886goes to zero at very late times independently of 119887

0

42 Cosmological Term Λ The corresponding Hamiltoniandensity becomes

H119868=

1

8

119890

minus(1205731+1205732+1205733)[minusΠ

2

1minus Π

2

2minus Π

2

3+ 2Π

1Π2

+ 2Π1Π3+ 2Π

2Π3minus

16

11

Π

2

120601+

6

11

Π120601Π120590minus

1

33

Π

2

120590

minus 16Λ119890

2(1205731+1205732+1205733)]

(29)

and corresponding Hamiltonrsquos equations are

Π

1015840

1= Π

1015840

2= Π

1015840

3= 4Λ119890

1205731+1205732+1205733

120573

1015840

1=

1

4

119890

minus(1205731+1205732+1205733)[minusΠ

1+ Π

2+ Π

3]

120573

1015840

2=

1

4

119890

minus(1205731+1205732+1205733)[minusΠ

2+ Π

1+ Π

3]

120573

1015840

3=

1

4

119890

minus(1205731+1205732+1205733)[minusΠ

3+ Π

1+ Π

2]

120601

1015840= 120601

0119890

minus(1205731+1205732+1205733)

120590

1015840= 120590

0119890

minus(1205731+1205732+1205733)

(30)

where the constants1206010and120590

0are the same as in previous case

To solve the last systemof equationswewill take the followingansatz

Π1= Π

2+ 120575

2= Π

3+ 120575

3 (31)

with 1205752and 120575

3constants By substituting into Hamiltonian

(29) we find a differential equation for the momentaΠ1given

by

1

Λ

Π

10158402

1minus 3Π

2

1+ ]Π

1+ 120575

1= 0 (32)

where the corresponding constants are

] = 2 (1205752+ 120575

3)

1205751(1199020) = (120575

2minus 120575

3)

2

minus 1199020

(33)

with the constant (related to the momenta scalar field)

1199020=

16

11

Π

2

120601minus

6

11

Π120601Π120590+

1

33

Π

2

120590 (34)

The solution for Π1is thus

Π1=

]6

+

1

6

radic]2 + 121205751cosh (radic3Λ120591) (35)

Hence using the last result (35) in the rest of equations in (30)we obtain that

1205731+ 120573

2+ 120573

3= ln(1

8

radic

]2 + 121205751

3Λ

sinh (radic3Λ120591)) (36)


and the corresponding solutions to the moduli fields are

120601 =

81206012

radic]2 + 121205751

ln(tanh(radic3Λ

2

120591))

120590 =

81205902

radic]2 + 121205751

ln(tanh(radic3Λ

2

120591))

(37)

With the last results we see that the solutions to the Misnervariables (Ω 120573

plusmn) are given by

Ω =

1

3

ln(18

radic

]2 + 121205751

3Λ

sinh (radic3Λ120591))

120573+=

2

3

(1205752minus 2120575

3)

radic]2 + 121205751

ln tanh(radic3Λ

2

120591)

120573minus= minus

2

radic3

1205752

radic]2 + 121205751

ln tanh(radic3Λ

2

120591)

(38)

and the isotropization parameter 120588119886is given by

120588119886=

Λ

3

(1205752minus 2120575

3)

2

+ 120575

2

2

]2 + 12 ((1205752minus 120575

3)

2

minus 1199020)

coth2 (radic3Λ

2

120591) (39)

Notice that the external four-dimensional volume expands astime 120591 runs and it is affectedwhether 119902

0is positive or negative

This in turn determines how fast the moduli fields evolve inrelation to each other (ie whether Π

120601is larger or smaller

than Π120590) Therefore isotropization is reached independently

of the values of 1199020but expansion of the Universe (and

shrinking of the internal volume) is affected by it

43 Matter Content and Cosmological Term For this casethe corresponding Hamiltonian density becomes (18) Sousing the Hamilton equation we can see that the momentaassociated with the scalars fields 120601 and 120590 are constants andwe will label these constants by 120601

1and 120590

1 respectively

Π

1015840

1= Π

1015840

2= Π

1015840

3

= 4Λ119890

1205731+1205732+1205733+ 16120587119866 (1 minus 120574) 120588

120574119890

minus120574(1205731+1205732+1205733)

120573

1015840

1=

1

4

119890

minus(1205731+1205732+1205733)[minusΠ

1+ Π

2+ Π

3]

120573

1015840

2=

1

4

119890

minus(1205731+1205732+1205733)[minusΠ

2+ Π

1+ Π

3]

120573

1015840

3=

1

4

119890

minus(1205731+1205732+1205733)[minusΠ

3+ Π

1+ Π

2]

120601

1015840= 120601

2119890

minus(1205731+1205732+1205733)

120590

1015840= 120590

2119890

minus(1205731+1205732+1205733)

(40)

Follow the same steps as in Section 42 we see that it ispossible to relate the momenta associated with 120573

1 120573

2 and 120573

3

in the following way

Π1= Π

2+ 119896

1= Π

3+ 119896

2 (41)

Also the differential equation for fields 120601 and 120590 can bereduced to quadrature as

120601 = 1206012int 119890

minus(1205731+1205732+1205733)119889120591

120590 = 1205902int 119890

minus(1205731+1205732+1205733)119889120591

(42)

We proceed to consider specific values for 120574

431 120574 = plusmn1 Cases Let us introduce the generic parameter

120582 =

4Λ 120574 = 1

4Λ + 32120587119866120588minus1 120574 = minus1

(43)

for whichΠ10158401= 120582119890

1205731+1205732+1205733 Substituting this intoHamiltonian

(18) we find the differential equation for the momenta Π1as

4

120582

Π

10158402

1minus 3Π

2

1+ ]Π

1+ 120575

1= 0 (44)

with the solution given by Π1is

Π1=

]6

+

1

6

radic]2 + 121205751cosh(

radic3120582

2

120591) (45)

On the other hand using the last result (45) we obtain fromHamilton equations that

1205731+ 120573

2+ 120573

3= ln(1

4

radic

1205753+ 12120575

1

3120582

sinh(radic3120582

2

120591)) (46)

The corresponding solutions to the moduli fields (42) can befound as

120601 =

81206012

radic]2 + 121205751

ln tanh(radic3120582

4

120591)

120590 =

81205902

radic]2 + 121205751


4

120591)

(47)


The solutions to Misner variables (16) (Ω 120573plusmn) are given by

Ω =

1

3

ln(14

radic

120575

2+ 12120575

1

3120582

sinh(radic3120582

2

120591))

120573+=

2

3

(1205752minus 2120575

3)

radic]2 + 121205751


2

120591)

+

1

6

ln(14

radic

]2 + 121205751

3120582

)

120573minus= minus

2

radic3

1205752

radic]2 + 121205751


2

120591)

minus

1

2radic3

ln(14

radic

]2 + 121205751

3120582

)

(48)

It is remarkable that (42) for the fields 120601 and 120590 aremaintainedfor all Bianchi Class A models and in particular when weuse the gauge119873 = 1198901205731+1205732+1205733 the solutions for these fields areindependent of the cosmological models whose solutions are120601 = (120601

08)119905 and 120590 = (120590

08)119905

After studying the 120574 = 0 case we will comment on thephysical significance of these solutions

432 120574 = 0 Case TheHamilton equations are

Π

1015840

1= Π

1015840

2= Π

1015840

3= 4Λ119890

1205731+1205732+1205733+ 120572

0 120572

0= 16120587119866120588

0

Π

1015840

120601= 0 Π

120601= 120601

1= 119888119905119890

(49)

from which the momenta Π1fulfil the equation

1

Λ

Π

10158402

1minus 3Π

2

1+ 120583Π

1+ 120585

1= 0 (50)

where

1205851= 120575

1minus

120572

2

0

Λ

(51)

The solution is given by

Π1=

]6

+

1

6


Using this result in the corresponding Hamilton equationswe obtain that

1205731+ 120573

2+ 120573

3= ln(1

8

radic

]2 + 121205851

3Λ

sinh (radic3Λ120591)) (53)

and the corresponding solutions to the dilaton field 120601 and themoduli field 120590 are given by

120601 =

81206012

radic]2 + 121205851

ln tanh(radic3Λ

2

120591)

120590 =

81205902

radic]2 + 121205851

ln tanh(radic3Λ

2

120591)

(54)

So the solutions to the Misner variables (Ω 120573plusmn) are

Ω =

1

3

ln(18

radic

]2 + 121205851

3Λ


120573+=

2

3

1205752minus 2120575

3

radic]2 + 121205851

ln tanh(radic3Λ

2

120591)

120573minus= minus

2

radic3

1205752

radic]2 + 121205851

ln tanh(radic3Λ

2

120591)

(55)

For both solutions 120574 = 0 plusmn1 we also observe that forpositive (negative) 119902

0 the Universe expands slower (faster)

than in the absence of extra dimensions while the internalspace shrinks faster (slower) for positive (negative) 119902

0 In this

case isotropization parameters are also accordingly affectedby 119902

0 through relations (34)

So far our analysis has been completely classic Now inthe next section we deal with the quantum scheme and weare going to solve the WDW equation in standard quantumcosmology

5 Quantum Scheme

Solutions to the Wheeler-DeWitt (WDW) equation dealingwith different problems have been extensively used in theliterature For example the important quest of finding atypical wave function for the Universe was nicely addressedin [39] while in [40] there appears an excellent summaryconcerning the problem of how a Universe emerged froma big bang singularity whihc cannot longer be neglectedin the GUT epoch On the other hand the best candidatesfor quantum solutions become those that have a dampingbehavior with respect to the scale factor represented in ourmodel with the Ω parameter in the sense that we obtain agood classical solution using the WKB approximation in anyscenario in the evolution of our Universe [41] The WDWequation for thismodel is achieved by replacing themomentaΠ119902120583 = minus119894120597

119902120583 associated with the Misner variables (Ω 120573

+ 120573

minus)

and the moduli fields (120601 120590) in Hamiltonian (15) The factor119890

minus3Ω may be factor ordered with ΠΩin many ways Hartle

and Hawking [41] have suggested what might be called asemigeneral factor ordering which in this case would order119890

minus3ΩΠ

2

Ωas

minus119890

minus(3minus119876)Ω120597Ω119890

minus119876Ω120597Ω= minus119890

minus3Ω120597

2

Ω+ 119876119890

minus3Ω120597Ω (56)

where 119876 is any real constant that measure the ambiguity inthe factor ordering in the variable Ω and the correspondingmomenta We will assume in the following this factor order-ing for the Wheeler-DeWitt equation which becomes

◻Ψ + 119876

120597Ψ

120597Ω

minus

48

11

120597

2Ψ

1205971206012minus

1

11

120597

2Ψ

1205971205902+

18

11

120597

2Ψ

120597120601120597120590

minus [119887120574119890

3(1minus120574)Ω+ 48Λ119890

6Ω]Ψ = 0

(57)


where ◻ is the three-dimensional drsquoLambertian in the ℓ120583 =(Ω 120573

+ 120573

minus) coordinates with signature (minus + +) On the other

hand we could interpret the WDW equation (57) as a time-reparametrization invariance of the wave function Ψ At aglance we can see that the WDW equation is static this canbe understood as the problem of time in standard quantumcosmology We can avoid this problem by measuring thephysical time with respect a kind of time variable anchoredwithin the system whichmeans that we could understand theWDW equation as a correlation between the physical timeand a fictitious time [42 43] When we introduce the ansatz

Ψ = Φ (120601 120590) 120595 (Ω 120573plusmn) (58)

in (57) we obtain the general set of differential equations(under the assumed factor ordering)

◻120595 + 119876

120597120595

120597Ω

minus [119887120574119890

3(1minus120574)Ω+ 48Λ119890

6Ωminus

120583

2

5

]120595 = 0 (59)

48

11

120597

2Φ

1205971206012+

1

11

120597

2Φ

1205971205902minus

18

11

120597

2Φ

120597120601120597120590

+ 120583

2Φ = 0 (60)

where we choose the separation constant 12058325 for conve-nience to reduce the second equation The solution to thehyperbolic partial differential equation (60) is given by

Φ(120601 120590) = 1198621sin (119862

3120601 + 119862

4120590 + 119862

5)

+ 1198622cos (119862

3120601 + 119862

4120590 + 119862

5)

(61)

where 119862119894

5

119894=1are integration constants and they are in

terms of 120583 We claim that this solution is the same for allBianchi Class A cosmological models because the Hamil-tonian operator in (57) can be written in separated way as119867(Ω 120573

plusmn 120601 120590)Ψ =

119867119892(Ω 120573

plusmn)Ψ +

119867119898(120601 120590)Ψ = 0 where

119867119892y 119867

119898represents the Hamiltonian to gravitational sector

and the moduli fields respectively To solve (59) we now set120595(Ω 120573

plusmn) = A(Ω)B

1(120573

+)B

2(120573

minus) obtaining the following set

of ordinary differential equations

119889

2A

119889Ω2minus 119876

119889A

119889Ω

+ [119887120574119890

minus3(120574minus1)Ω+ 48Λ119890

6Ω+ 120590

2]A = 0

119889

2B1

1198891205732

+

+ 119886

2

2B

1= 0 997904rArr

B1= 120578

1119890

1198941198862120573++ 120578

2119890

minus1198941198862120573+

119889

2B2

1198891205732

minus

+ 119886

2

3B

2= 0 997904rArr

B2= 119888

1119890

1198941198863120573minus+ 119888

2119890

minus1198941198863120573minus

(62)

where 1205902 = 11988621+ 119886

2

2+ 119886

2

3and 119888

119894and 120578

119894are constants We now

focus on the Ω dependent part of the WDW equation Wesolve this equation when Λ = 0 and Λ = 0

(1) For the casewith null cosmological constant and 120574 = 1

119889

2A

119889Ω2minus 119876

119889A

119889Ω

+ [119887120574119890

minus3(1minus120574)Ω+ 120590

2]A = 0 (63)

and by using the change of variable

119911 =

radic119887120574

119901

119890

minus(32)(120574minus1)Ω

(64)

where 119901 is a parameter to be determined we have

119889A

119889Ω

=

119889A

119889119911

119889119911

119889Ω

= minus

3

2

(120574 minus 1) 119911

119889A

119889119911

119889

2A

119889Ω2=

9

4

(120574 minus 1)

2

119911

2 1198892A

1198891199112+

9

4

(120574 minus 1)

2

119911

119889A

119889119911

(65)

Hence we arrive at the equation

9

4

(120574 minus 1)

2

119911

2 1198892A

1198891199112+

9

4

(120574 minus 1)

2

119911

119889A

119889119911

minus

3

2

119876 (120574 minus 1) 119911

119889A

119889119911

+ [119901

2119911

2+ 120590

2] = 0

(66)

where we have assumed thatA is of the form [44]

A = 119911

119902119876Φ (119911) (67)

with 119902 being yet to be determined We thus get aftersubstituting in (66)

9

4

(120574 minus 1)

2

119911

119902119876[119911

2 1198892Φ

1198891199112+ 119911(1 + 2119902119876

+

2

3

119876

120574 minus 1

)

119889Φ

119889119911

+ (

4119901

2

9 (120574 minus 1)

2119911

2

+ 119876

2119902

2+

2

3

119902

120574 minus 1

+

4120590

2

9 (120574 minus 1)

2)Φ] = 0

(68)

which can be written as

119911

2 1198892Φ

1198891199112+ 119911

119889Φ

119889119911

+ [119911

2minus

1

9 (120574 minus 1)

2(119876

2minus 4120590

2)]Φ

= 0

(69)

which is the Bessel differential equation for the function Φwhen 119901 and 119902 are fixed to

119902 = minus

1

3 (120574 minus 1)

119901 =

3

2

1003816100381610038161003816120574 minus 1

1003816100381610038161003816

(70)

which in turn means that transformations (64) and (67) are

119911 =

2radic119887120574

3

1003816100381610038161003816120574 minus 1

1003816100381610038161003816

119890


A = 119911

minus(1198763(120574minus1))Φ(119911)

(71)


Hence the solution for (63) is of the form

A120574= 119888

120574(

2radic119887120574

3

1003816100381610038161003816120574 minus 1

1003816100381610038161003816

119890

minus(32)(120574minus1)Ω)

minus1198763(120574minus1)

sdot 119885119894](

2radic119887120574

3

1003816100381610038161003816120574 minus 1

1003816100381610038161003816

119890


(72)

with

] = plusmnradic1

9 (120574 minus 1)

2(4120590

2minus 119876

2) (73)

where119885119894] = 119869119894] is the ordinaryBessel functionwith imaginary

order For the particular case when the factor ordering is zerowe can easily construct a wave packet [41 45 46] using theidentity

int

infin

minusinfin

sech(120587120578

2

) 119869119894120578(119911) 119889120578 = 2 sin (119911) (74)

so the total wave function becomes

Ψ (Ω 120573plusmn 120601 120590) = Φ (120601 120590)

sdot sin(2radic119887120574

3

1003816100381610038161003816120574 minus 1

1003816100381610038161003816

119890


sdot [1205781119890

1198941198862120573++ 120578

2119890

minus1198941198862120573+]

sdot [1198881119890

1198941198863120573minus+ 119888

2119890

minus1198941198863120573minus]

(75)

Notice that the influence of extra dimensions through thepresence of the moduli 120601 and 120590 appears in the solution inthe amplitude of the wave functionWe will comment on thislater on

(2) Case with null cosmological constant and 120574 = 1 Forthis case we have

119889

2A1

119889Ω2minus 119876

119889A1

119889Ω

+ 120590

2

1A1= 0 120590

2

1= 119887

1+ 120590

2

(76)

whose solution is

A1= 119860

1119890

((119876+radic1198762+41205902

1)2)Ω

+ 1198602119890

((119876minusradic1198762+41205902

1)2)Ω

(77)

So the total wave function becomes

Ψ (Ω 120573plusmn 120601 120590) = Φ (120601 120590)

sdot [1198601119890

((119876+radic1198762+41205902

1)2)Ω

+ 1198602119890

((119876minusradic1198762+41205902

1)2)Ω

]

sdot [1205781119890

1198941198862120573++ 120578

2119890

minus1198941198862120573+] [119888

1119890

1198941198863120573minus+ 119888

2119890

minus1198941198863120573minus]

(78)

(3) If we include the cosmological constant term for thisparticular case we have [44]

A = 119890

(1198762)Ω119885] (4radic

Λ

3

119890

3Ω) ] = plusmn

1

6

radic1198762+ 4120590

2

1 (79)

where Λ gt 0 in order to have ordinary Bessel functionsas solutions in other case we will have the modified Besselfunction When the factor ordering parameter119876 equals zerowe have the same generic Bessel functions as solutionshaving the imaginary order ] = plusmn119894radic1205902

13

(4) 120574 = minus1 and Λ = 0 and any factor ordering 119876

119889

2Aminus1

119889Ω2minus 119876

119889Aminus1

119889Ω

+ [119889minus1119890

6Ωminus 120590

2]A

minus1= 0

119889minus1= 48120583

minus1+ 48Λ

(80)

with the solution

Aminus1= (

radic119889minus1

3

119890

3Ω)

1198766

119885] (radic119889

minus1

3

119890

3Ω)

] = plusmn1

6

radic1198762+ 4120590

2

(81)

where119885] is a generic Bessel function When 119887minus1gt 0 we have

ordinary Bessel functions in other case we have modifiedBessel functions

(5) 120574 = 0 Λ = 0 and factor ordering 119876 = 0

119889

2A0

119889Ω2+ (48Λ119890

6Ω+ 119887

0119890

3Ω+ 120590

2)A

0= 0 119887

0= 48120583

0(82)

with the solution

A0= 119890

minus3Ω2[119863

1119872

minus119894119887024radic3Λminus1198941205903

(

8119894119890

3ΩradicΛ

radic3

)

+ 1198632119882minus119894119887024radic3Λminus1198941205903

(

8119894119890

3ΩradicΛ

radic3

)]

(83)

where119872119896119901

And119882119896119901

are Whittaker functions and119863119894are in-

tegration constants

6 Final Remarks

In this work we have explored a compactification of a ten-dimensional gravity theory coupled with a time-dependentdilaton into a time-dependent six-dimensional torus Theeffective theory which emerges through this process resem-bles the Einstein frame to that described by the BianchiI model By incorporating the barotropic matter and cos-mological content and by using the analytical procedureof Hamilton equation of classical mechanics in appropri-ate coordinates we found the classical solution for theanisotropic Bianchi type I cosmologicalmodels In particularthe Bianchi type I is completely solved without using aparticular gauge With these solutions we can validate ourqualitative analysis on isotropization of the cosmologicalmodel implying that the volume becomes larger in thecorresponding time evolution

We find that for all cases the Universe expansion isgathered as time runs while the internal space shrinksQualitatively this model shows us that extra dimensions are


forced to decrease its volume for an expanding UniverseAlso we notice that the presence of extra dimensions affectshow fast the Universe (with matter) expands through thepresence of a constant related to the moduli momenta Thisis not unexpected since we are not considering a potentialfor the moduli which implies that they are not stabilizedand consequently an effective model should only take intoaccount their constant momentaWe observe that isotropiza-tion is not affected in the cases without matter but in thematter presence isotropization can be favored or retardedaccording to how fast the moduli evolve with respect to eachother

It could be interesting to study other types of matterin this context beyond the barotropic matter For instancethe Chaplygin gas with a proper time characteristic to thismatter leads to the presence of singularities types I IIIII and IV (generalizations to these models are presentedin [47ndash49]) which also appear in the phantom scenario todarkenergy matter In our case since the matter we areconsidering is barotropic initial singularities of these typesdo not emerge from our analysis implying also that phantomfields are absent It is important to remark that this model isa very simplified one in the sense that we do not considerthe presence of moduli-dependent matter and we do notanalyze under which conditions inflation is present or howit starts We plan to study this important task in a differentwork

Concerning the quantum schemewe can observe that thisanisotropic model is completely integrable without employ-ing numerical methods similar solutions to partial differ-ential equation into the gravitational variables have beenfound in [50] and we obtain that the solutions in the modulifields are the same for all Bianchi Class A cosmologicalmodels because the Hamiltonian operator in (57) can bewritten in separated way as 119867(Ω 120573

plusmn 120601 120590)Ψ =

119867119892(Ω 120573

plusmn)Ψ +

119867119898(120601 120590)Ψ = 0 where 119867

119892y 119867

119898represents the Hamiltonian

to gravitational sector and themoduli fields respectively withthe full wave function given byΨ = Φ(120601 120590)Θ(Ω 120573

plusmn) In order

to have the best candidates for quantum solutions becomethose that have a damping behavior with respect to the scalefactor represented in ourmodelwithΩparameter in thiswaywe will drop in the full solution the modified Bessel function119868](119911) or Bessel function 119884](119911) with ] being the order of thesefunctions

We can observe that the quantum solution in the Ωsector is similar to the corresponding FRW cosmologicalmodel found in different schemes [12 22 23 32 43] Alsosimilar analysis based on a Bianchi type II model can befound in recent published paper by two authors of this work[51]

The presence of extra dimensions could have a strongerinfluence on the isotropization process by having a moduli-dependent potential which can be gathered by turning onextra fields in the internal space called fluxes It will beinteresting to consider a more complete compactificationprocess inwhich allmoduli are considered as time-dependentfields as well as time-dependent fluxes We leave theseimportant analyses for future work

Appendix

Dimensional Reduction

With the purpose to be consistent here we present the mainideas to dimensionally reduce action (1) By the use of theconformal transformation

119866119872119873

= 119890

Φ2119866

119864

119872119873 (A1)

action (1) can be written as

119878 =

1

21205812

10

int119889

10119883radicminus119866(119890

Φ2R + 4119866

119864119872119873nabla119872Φnabla

119873Φ)

+ int119889

10119883radicminus119866

119864119890

5Φ2Lmatt

(A2)

where the ten-dimensional scalar curvature R transformsaccording to the conformal transformation as

R = 119890

minusΦ2(R

119864minus

9

2

119866

119864119872119873nabla119872nabla119873Φ

minus

9

2

119866

119864119872119873nabla119872Φnabla

119873Φ)

(A3)

By substituting expression (A3) in (A2) we obtain

119878 =

1

21205812

10

int119889


119864(R

119864minus

9

2

119866

119864119872119873nabla119872nabla119873Φ

minus

1

2

119866

119864119872119873nabla119872Φnabla

119873Φ) + int119889


119864119890

5Φ2Lmatt

(A4)

The last expression is the ten-dimensional action in theEinstein frame Expressing the metric determinant in four-dimensions in terms of the moduli field 120590 we have

det119866119872119873

=119866 = 119890

minus12120590 (A5)

By substituting the last expression (A5) in (1) andconsidering that

R(10)

=R(4)

minus 42

120583]nabla120583120590nabla]120590 + 12

120583]nabla120583nabla]120590 (A6)

we obtain that

119878 =

1

21205812

10

int119889

4119909119889


minus6120590119890

minus2Φ[R

(4)

minus 42

120583]nabla120583120590nabla]120590 + 12

120583]nabla120583nabla]120590 + 4

120583]nabla120583Φnabla]Φ]

(A7)

Let us redefine in the last expression the dilaton field Φ as

Φ = 120601 minus

1

2

ln (119881) (A8)

where 119881 = int1198896119910 So expression (A7) can be written as

119878 =

1

21205812

10

int119889


minus2(120601+3120590)[R

(4)

minus 42

120583]nabla120583120590nabla]120590

+ 12

120583]nabla120583nabla]120590 + 4

120583]nabla120583120601nabla]120601]

(A9)


The four-dimensional metric 120583] means that the metric

is in the string frame and we should take a conformaltransformation linking the String and Einstein frames Welabel by 119892

120583] the metric of the external space in the Einsteinframe this conformal transformation is given by

120583] = 119890

2Θ119892120583] (A10)

which after some algebra gives the four-dimensional scalarcurvature

R(4)

= 119890

minus2Θ(R

(4)minus 6119892

120583]nabla120583nabla]Θ minus 6119892

120583]nabla120583Θnabla]Θ) (A11)

where the function Θ is given by the transformation

Θ = 120601 + 3120590 + ln(120581

2

10

1205812

4

) (A12)

Now wemust replace expressions (A10) (A11) and (A12) inexpression (A9) and we find

119878 =

1

21205812

4

int119889



120583]nabla120583nabla]120590

minus 2119892



120583]nabla120583120601nabla]120590)

(A13)

At first glance it is important to clarify one point related tothe stress-energy tensor which has the matrix form (6) thistensor belongs to the String frame In order towrite the stress-energy tensor in the Einstein frame we need to work with thelast integrand of expression (1) So after taking the variationwith respect to the ten-dimensional metric and opening theexpression we see that

int119889

10119883radicminus119866119890

minus2Φ120581

2

10119890

2Φ119879119872119873119866

119872119873

= int119889

4119909radicminus119892119890

2(Θ+120601)119879119872119873119866

119872119873

= int119889

4119909radicminus119892 (119890

2120601119879120583]119892

120583]+ 119890

2(Θ+120601)119879119898119899

119898119899)

(A14)

where we can observe that the four-dimensional stress-energy tensor in the Einstein frame is defined as we said inexpression (7)

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was partially supported by CONACYT 167335179881 Grants and PROMEPGrants UGTO-CA-3 andDAIP-UG6402015Thiswork is part of the collaborationwithin theInstituto Avanzado de Cosmologıa and Red PROMEP Grav-itation and Mathematical Physics under project QuantumAspects of Gravity in Cosmological Models Phenomenologyand Geometry of Spacetime One of authors (L ToledoSesma) was supported by a PhD scholarship in the graduateprogram by CONACYT

References

[1] P A R Ade N Aghanim M Arnaud et al ldquoPlanck 2015results XVIII Background geometry and topologyrdquo httparxivorgabs150201593

[2] E Russell C B Kilinc and O K Pashaev ldquoBianchi I modelan alternative way to model the present-day UniverserdquoMonthlyNotices of the Royal Astronomical Society vol 442 no 3 pp2331ndash2341 2014

[3] T Damour and A M Polyakov ldquoThe string dilaton and a leastcoupling principlerdquoNuclear Physics B vol 423 no 2-3 pp 532ndash558 1994

[4] J H Horne and G W Moore ldquoChaotic coupling constantsrdquoNuclear Physics B vol 432 no 1-2 pp 109ndash126 1994

[5] T Banks M Berkooz S H Shenker G W Moore and P JSteinhardt ldquoModular cosmologyrdquoPhysical ReviewD vol 52 no6 pp 3548ndash3562 1995

[6] T Damour M Henneaux B Julia and H Nicolai ldquoHyperbolicKac-Moody algebras and chaos in KALuza-KLEin modelsrdquoPhysics Letters B vol 509 no 3-4 pp 323ndash330 2001

[7] T Banks W Fischler and L Motl ldquoDualities versus singulari-tiesrdquo Journal of High Energy Physics vol 1999 no 1 article 0191999

[8] R Kallosh L Kofman and A Linde ldquoPyrotechnic universerdquoPhysical Review D vol 64 no 12 Article ID 123523 18 pages2001

[9] R Kallosh L Kofman A Linde andA Tseytlin ldquoBPS branes incosmologyrdquo Physical ReviewD vol 64 no 12 Article ID 1235242001

[10] D Baumann and L McAllister Inflation and String TheoryCambridge Monographs on Mathematical Physics CambridgeUniversity Press New York NY USA 2015

[11] J Khoury B A Ovrut N Seiberg P J Steinhardt andN TurokldquoFrom big crunch to big bangrdquo Physical Review D vol 65 no 8Article ID 086007 2002

[12] C Martınez-Prieto O Obregon and J Socorro ldquoClassical andquantum time dependent solutions in string theoryrdquo Interna-tional Journal of Modern Physics A vol 19 no 32 pp 5651ndash56612004

[13] J Khoury B A Ovrut P J Steinhardt andN Turok ldquoEkpyroticuniverse colliding branes and the origin of the hot big bangrdquoPhysical Review D vol 64 no 12 Article ID 123522 24 pages2001

[14] K Enqvist E Keski-Vakkuri and S Rasanen ldquoHubble law andbrane matter after ekpyrosisrdquo Nuclear Physics B vol 64 p 3882001

[15] G Veneziano ldquoScale factor duality for classical and quantumstringsrdquo Physics Letters B vol 265 no 3-4 pp 287ndash294 1991

[16] M Gasperini and G Veneziano ldquoPre-big-bang in string cos-mologyrdquo Astroparticle Physics vol 1 no 3 pp 317ndash339 1993

[17] C J Isham ldquoCanonical quantum gravity and the problem oftimerdquo in Integrable Systems Quantum Groups and QuantumFieldTheories L A Ibort andM A Rodrıguez Eds vol 409 ofNATO ASI Series pp 157ndash287 1993

[18] K V Kuchar ldquoTime and interpretations of quantum gravityrdquoInternational Journal of Modern Physics D vol 20 no 3 pp 3ndash86 2011

[19] A P Billyard A A Coley and J E Lidsey ldquoQualitative analysisof string cosmologiesrdquo Physical Review D vol 59 no 12 ArticleID 123505 7 pages 1999


[20] A P Billyard A A Coley and J E Lidsey ldquoQualitative analysisof early universe cosmologiesrdquo Journal of Mathematical Physicsvol 40 no 10 pp 5092ndash5105 1999

[21] A P Billyard A A Coley J E Lidsey and U S NilssonldquoDynamics of M-theory cosmologyrdquo Physical Review D vol 61no 4 Article ID 043504 2000

[22] M Cavaglia and P V Moniz ldquoCanonical and quantum FRWcosmological solutions in M-theoryrdquo Classical and QuantumGravity vol 18 no 1 pp 95ndash120 2001

[23] M Cavaglia and P VMoniz ldquoFRWcosmological solutions inMtheoryrdquo inRecent Developments inTheoretical and ExperimentalGeneral Relativity Gravitation and Relativistic Field TheoriesProceedings 9th Marcel Grossmann Meeting MGrsquo9 Rome ItalyJuly 2ndash8 2000 Pts A-C 2000

[24] C Brans and R H Dicke ldquoMachrsquos principle and a relativistictheory of gravitationrdquo Aps Journals Archive vol 124 no 3 p925 1961

[25] D H Wesley ldquoOxidised cosmic accelerationrdquo Journal of Cos-mology and Astroparticle Physics vol 2009 article 041 2009

[26] J A Casas J Garcia-Bellido and M Quiros ldquoOn the gravitytheories and cosmology from stringsrdquo Nuclear Physics B vol361 no 3 pp 713ndash728 1991

[27] E J Copeland A Lahiri and D Wands ldquoLow energy effectivestring cosmologyrdquo Physical Review D vol 50 no 8 pp 4868ndash4880 1994

[28] J E Lidsey DWands and E J Copeland ldquoSuperstring cosmol-ogyrdquo Physics Reports vol 337 no 4-5 pp 343ndash492 2000

[29] M Gasperini Elements of String Cosmology Cambridge Uni-versity Press New York NY USA 2007

[30] S M M Rasouli M Farhoudi and P V Moniz ldquoModifiedBrans-Dicke theory in arbitrary dimensionsrdquo Classical andQuantum Gravity vol 31 no 11 Article ID 115002 26 pages2014

[31] M P Ryan and L C Shepley Homogeneous Relativistic Cos-mologies Princeton University Press Princeton NJ USA 1975

[32] J Socorro ldquoClassical solutions from quantum regime forbarotropic FRW modelrdquo International Journal of TheoreticalPhysics vol 42 no 9 pp 2087ndash2096 2003

[33] B D Chowdhury and S D Mathur ldquoFractional brane state inthe early Universerdquo Classical and Quantum Gravity vol 24 no10 pp 2689ndash2720 2007

[34] S Nojiri O Obregon S D Odintsov and K E Osetrin ldquoIn-ducedwormholes due to quantum effects of spherically reducedmatter in largeN approximationrdquo Physics Letters B vol 449 no3-4 pp 173ndash179 1999

[35] S Nojiri O Obregon S D Odintsov and K E Osetrin ldquo(Non)singular Kantowski-Sachs universe from quantum sphericallyreduced matterrdquo Physical Review D vol 60 no 2 Article ID024008 1999

[36] S Nojiri O Obregon S D Odintsov and K Osetrinldquo(Non)singular Kantowski-Sachs universe from quantumspherically reduced matterrdquo Physical Review D vol 60 no 2Article ID 024008 1999

[37] E Martinez-Gonzalez and J L Sanz ldquoΔ TT and the isotropyof the Universerdquo Astronomy and Astrophysics vol 300 p 3461995

[38] J Socorro L O Pimentel and A Espinoza-Garcıa ldquoClassicalBianchi type I cosmology in K-essence theoryrdquo Advances inHigh Energy Physics vol 2014 Article ID 805164 11 pages 2014

[39] G W Gibbons and L P Grishchuk ldquoWhat is a typical wavefunction for the universerdquoNuclear Physics B vol 313 no 3 pp736ndash748 1989

[40] S Gottlber L Z Fang and R Ruffini Eds Quantum Cosmol-ogy vol 3 of Advanced Series in Astrophysics and CosmologyWorld Scientific Singapore 1987

[41] J B Hartle and SW Hawking ldquoWave function of the universerdquoPhysical Review D vol 28 no 12 pp 2960ndash2975 1983

[42] J H Bae ldquoMixmaster revisited wormhole solutions to theBianchi IX Wheeler-DeWitt equation using the Euclidean-signature semi-classical methodrdquo Classical and Quantum Grav-ity vol 32 no 7 Article ID 075006 15 pages 2015

[43] P V MonizQuantum Cosmology The Supersymmetric Perspec-tive Fundamentals vol 1 Springer 2010

[44] V F Zaitsev and A D Polyanin Handbook of Exact Solutionsfor Ordinary Differential Equations Taylor amp Francis 2002

[45] C Kiefer ldquoWave packets in quantum cosmology and thecosmological constantrdquo Nuclear Physics B vol 341 no 1 pp273ndash293 1990

[46] C Kiefer ldquoWave packets inminisuperspacerdquo Physical Review DParticles and Fields vol 38 no 6 pp 1761ndash1772 1988

[47] K Bamba J de Haro and S D Odintsov ldquoFuture singularitiesand teleparallelism in loop quantum cosmologyrdquo Journal ofCosmology and Astroparticle Physics vol 2013 article 008 2013

[48] S Nojiri S D Odintsov and S Tsujikawa ldquoProperties ofsingularities in the (phantom) dark energy universerdquo PhysicalReview D vol 71 no 6 Article ID 063004 2005

[49] L Fernandez-Jambrina ldquo120596-cosmological singularitiesrdquoPhysicalReview D vol 82 no 12 Article ID 124004 2010

[50] J Socorro M Sabido M A Sanchez and M G Frıas PalosldquoAnisotropic cosmology in Saez-Ballester theory classical andquantum solutionsrdquo Revista Mexicana de Fısica vol 56 no 2pp 166ndash171 2010

[51] J Socorro and L Toledo Sesma ldquoTime-dependent toroidalcompactification proposals and the Bianchi type II modelclassical and quantum solutionsrdquoTheEuropean Physical JournalPlus vol 131 no 3 article 71 10 pages 2016

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



AstronomyAdvances in

International Journal of


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Statistical MechanicsInternational Journal of


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Solid State PhysicsJournal of

Computational Methods in Physics

Journal of



Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014


PhotonicsJournal of


Journal of

Biophysics


ThermodynamicsJournal of


modulos (for instance by assuming that it is already stabilizedby the presence of a string field in higher scales) This willallow us to construct classical effective models with twomoduli

However we are also interested in possible quantumaspects of our model Quantum implications on cosmologyfrommore fundamental theories are expected due to differentobservations For instance it has been pointed out thatthe presence of extra dimensions leads to an interestingconnection to the ekpyrotic model [13] which generatedconsiderable activity [8 9 14] The essential ingredient inthesemodels (see for instance [11]) is to consider an effectiveaction with a graviton and a massless scalar field the dilatondescribing the evolution of the Universe while incorporatingsome of the ideas of pre-big-bang proposal [15 16] in thatthe evolution of the Universe began in the far past Onthe other hand it is well known that relativistic theoriesof gravity such as general relativity or string theories areinvariant under reparametrization of time Quantization ofsuch theories presents a number of problems of principleknown as ldquothe problem of timerdquo [17 18] This problem ispresent in all systems whose classical version is invariantunder time reparametrization leading to its absence at thequantum level Therefore the formal question involves howto handle the classical Hamiltonian constraintH asymp 0 in thequantum theory Also connected with the problem of time isthe ldquoHilbert space problemrdquo [17 18] referring to the not at allobvious selection of the inner product of states in quantumgravity and whether there is a need for such a structure at allThe above features as it is well known point out the necessityto construct a consistent theory of gravity at quantum level

Analyses of effective four-dimensional cosmologies de-rived from M-theory and Type IIA string theory wereconsidered in [19ndash21] where string fluxes are related to thedynamical behavior of the solutions A quantum descriptionof the model was studied in [22 23] where a flat Friedmann-Robertson-Walker geometry was considered for the extendedfour-dimensional space-time while a geometry given by 1198781 times119879

6 was assumed for the internal seven-dimensional spacehowever the dynamical fields are in the quintom schemesince one of the fields has a negative energy Under a similarperspective we study the Hilbert space of quantum states ona Bianchi I geometry with two time-dependent scalar moduliderived from a ten-dimensional effective action containingthe dilaton and the Kahler parameter from a six-dimensionaltorus compactification We find that in this case the wavefunction of this Universe is represented by two factors onedepending onmoduli and the second one depending on grav-itational fieldsThis behavior is a property of all cosmologicalBianchi Class A models

The work is organized in the following form In Section 2we present the construction of our effective action by com-pactification on a time-dependent torus while in Section 3we study its Lagrangian and Hamiltonian descriptions usingas toymodel the Bianchi type I cosmologicalmodel Section 4is devoted to finding the corresponding classical solutionsfor few different cases involving the presence or absence ofmatter content and the cosmological constant term Using

the classical solutions found in previous sections we presentan isotropization mechanism for the Bianchi I cosmolog-ical model through the analysis of the ratio between theanisotropic parameters and the volume of the Universeshowing that in all the cases we have studied its valuekeeps constant or runs into zero for late times In Section 5we present some solutions to the corresponding Wheeler-DeWitt (WDW) equation in the context of standard quantumcosmology and finally our conclusions are presented inSection 6

2 Effective Model

We start from a ten-dimensional action coupled with adilaton (which is the bosonic component common to allsuperstring theories) which after dimensional reduction canbe interpreted as a Brans-Dicke like theory [24] In thestring frame the effective action depends on two spacetime-dependent scalar fields the dilaton Φ(119909120583) and the Kahlermodulus 120590(119909120583) For simplicity in this work we will assumethat these fields only depend on time The high-dimensional(effective) theory is therefore given by

119878 =

1

21205812

10

int119889

10119883radicminus119866119890

minus2Φ[R

(10)

+ 4119866

119872119873

nabla119872Φnabla

119873Φ]

+ int119889

10119883radicminus119866L

119898

(1)

where all quantities refer to the string frame while the ten-dimensional metric is described by

119889119904

2=119866119872119873119889119883

119872119889119883

119873=

120583]119889119909120583119889119909

]+ ℎ

119898119899119889119910

119898119889119910

119899 (2)

where119872119873 are the indices of the ten-dimensional spaceand Greek indices 120583 ] = 0 3 and Latin indices119898 119899 = 4 9 correspond to the external and internalspaces respectively We will assume that the six-dimensionalinternal space has the form of a torus with a metric given by

ℎ119898119899= 119890

minus2120590(119905)120575119898119899 (3)

with 120590 being a real parameterDimensionally reducing the first term in (1) to four-

dimensions in the Einstein frame (see the Appendix fordetails) gives

1198784=

1

21205812

4

int119889


120583]nabla120583120601nabla]120601

minus 96119892


120583]nabla120583120601nabla]120590)

(4)

where 120601 = Φ + (12) ln(119881) with 119881 given by

119881 = 119890

6120590(119905)Vol (1198836) = int119889

6119910 (5)

By considering only a time-dependence on the moduli onecan notice that for the internal volume Vol(119883

6) to be small

the modulus 120590(119905) should be a monotonic increasing function





119879119872119873

= (

119879120583] 0

0119879119898119899

) (6)

where 119879120583] and 119879


components of 119879119872119873





119879120583] = 119890

2120601119879120583] (7)





119889119904

2= minus119873 (119905) 119889119905

2+ 119890

2Ω(119905)(119890

2120573(119905))

119894119895120596

119894120596

119895 (8)


matrix 120573119894119895= diag(120573

++radic3120573

minus 120573

+minusradic3120573

minus minus2120573

+)Ω(119905) and 120573

plusmn


119894= (12)119862

119894

119895119896120596

119895and 120596

119896 with119862

119894



119889119904

2

119868= minus119873

2119889119905

2+ 119877

2

1119889119909

2+ 119877

2

2119889119910

2+ 119877

2

3119889119911

2 (9)

where 119877119894

3


1198771= 119890

Ω+120573++radic3120573

minus

1198772= 119890


minus

1198773= 119890

Ωminus2120573+

119881 = 119877111987721198773= 119890

3Ω

(10)



119879120583] = (119901 + 120588) 119906120583119906] + 119901119892120583] (11)






= 16119873120587119866119873119862120574119890

minus3(1+120574)Ω+ 2119873Λ119890

3Ω

(12)


L119868=

119890

3Ω

119873

[6Ω

2

minus 6120573

2

+minus 6

120573

2

minus+ 96

2+ 36

120601 + 2

120601

2

+ 16120587119866119873

2120588 + 2Λ119873

2]

(13)


120597L120597120583 where 119902120583 = (Ω 120573+ 120573


ΠΩ=

12119890

3ΩΩ

119873

Ω =

119873119890

minus3ΩΠΩ

12

Πplusmn= minus

12119890

3Ω120573plusmn

119873

120573plusmn= minus

119873119890


12

Π120601=

119890

3Ω

119873

[36 + 4120601]

120601 =

119873119890

minus3Ω

44

[3Π120590minus 16Π

120601]

Π120590=

119890

3Ω

119873

[192 + 36120601] =

119873119890

minus3Ω

132

[9Π120601minus Π

120590]

(14)







H119868=

119890

minus3Ω

24

[Π

2

Ωminus Π

2

+minus Π

2

minusminus

48

11

Π

2

120601+

18

11

Π120601Π120590

minus

1

11

Π

2

120590minus 384120587119866

119873120588120574119890

3(1minus120574)Ωminus 48Λ119890

6Ω]

(15)




1205731= Ω + 120573

++radic3120573

minus

1205732= Ω + 120573

+minusradic3120573

minus

1205733= Ω minus 2120573

+

(16)


L119868=

119890

1205731+1205732+1205733

119873

(2120601

2

+ 36120601 + 96

2+ 2

1205731

1205732

+ 21205731

1205733+ 2

1205733

1205732+ 2Λ119873

2

+ 16120587119866119873

2120588120574119890

minus(1+120574)(1205731+1205732+1205733))

(17)


H119868=

1

8

119890

minus(1205731+1205732+1205733)[minusΠ

2

1minus Π

2

2minus Π

2

3+ 2Π

1Π2

+ 2Π1Π3+ 2Π

2Π3minus

16

11

Π

2

120601+

6

11

Π120601Π120590minus

1

33

Π

2

120590

minus 128120587119866120588120574119890

(1minus120574)(1205731+1205732+1205733)minus 16Λ119890

2(1205731+1205732+1205733)]

(18)



Ω

10158402= 120573

10158402

++ 120573

10158402

minus+ 16120590

10158402+

1

3

120601

10158402+ 6120601

1015840120590

1015840

+

8

3

120587119866120583120574119890

minus3(1+120574)Ω+

Λ

3

(19)




++120573

10158402

minus)119867

2rarr 0

(120573

10158402

++120573

10158402




120588119886= 120573

10158402

++ 120573

10158402

minus (20)




120601= 119901

120601= constant

Π120590= 0 997888rarr Π

120590= 119901

120590= constant

Π+= 0 997888rarr Π

+= 119901

+= constant


minus= 119901

minus= constant

(21)


Ωas 1205812

Ωsim 119901

2

++119901

2

minus+(16132

2)(119901

120590minus

9119901120601)

2+(1(3sdot44

2))(3119901

120590minus16119901

120601)

2+(1(22sdot44))(119901

120590minus9119901

120601)(3119901

120590minus

16119901120601) we have that

120588119886sim 119890

minus6Ω

120588120601120590sim 119890

minus6Ω

Ω

10158402sim

Λ

3

+ 120581Ω

2119890

minus6Ω+ 119887

120574119890

minus3(1+120574)Ω

119887120574=

8

3

120587119866119873120583120574

(22)


120588120601120590

sim constant

120588119886

120588120574

sim 119890

3Ω(120574minus1)

120588119886

Ω10158402sim

1

120581Ω

2+ (Λ3) 119890

6Ω+ 119887

1205741198903(1minus120574)Ω

(23)


4 Case of Interest





that

H119868vac=

1

8

119890

minus(1205731+1205732+1205733)[minusΠ

2

1minus Π

2

2minus Π

2

3+ 2Π

1Π2

+ 2Π1Π3+ 2Π

2Π3minus

16

11

Π

2

120601+

6

11

Π120601Π120590minus

1

33

Π

2

120590]

(24)



119875119894= 120597H120597119902

119894

become


119894= constant

Π120601= 0 997904rArr Π

120601= constant

Π120590= 0 997904rArr Π

120590= constant

120573

1015840

1=

1

4

119890

minus(1205731+1205732+1205733)[minusΠ

1+ Π

2+ Π

3]

120573

1015840

2=

1

4

119890

minus(1205731+1205732+1205733)[minusΠ

2+ Π

1+ Π

3]

120573

1015840

3=

1

4

119890

minus(1205731+1205732+1205733)[minusΠ

3+ Π

1+ Π

2]

120601

1015840= 120601

0119890

minus(1205731+1205732+1205733)

120590

1015840= 120590

0119890

minus(1205731+1205732+1205733)

(25)



1205731+ 120573

2+ 120573

3= ln [120598120591 + 119887

0] 120598 = 119887

1+ 119887

2+ 119887

3 (26)

with 119887119894= Π

119894and 119887



Ω = ln (120598120591 + 1198870)

13

120573+= ln (120598120591 + 119887

0)

(120598minus31198873)6120598

120573minus= ln (120598120591 + 119887

0)

(1198873+21198871minus120598)2radic3120598

(27)


120601 =

1206010

120598

ln (120598120591 + 1198870)

120590 =

1205900

120598

ln (120598120591 + 1198870)

(28)

where 1206010= Π

120601and 120590

0= Π




0


H119868=

1

8

119890

minus(1205731+1205732+1205733)[minusΠ

2

1minus Π

2

2minus Π

2

3+ 2Π

1Π2

+ 2Π1Π3+ 2Π

2Π3minus

16

11

Π

2

120601+

6

11

Π120601Π120590minus

1

33

Π

2

120590

minus 16Λ119890

2(1205731+1205732+1205733)]

(29)


Π

1015840

1= Π

1015840

2= Π

1015840

3= 4Λ119890

1205731+1205732+1205733

120573

1015840

1=

1

4

119890

minus(1205731+1205732+1205733)[minusΠ

1+ Π

2+ Π

3]

120573

1015840

2=

1

4

119890

minus(1205731+1205732+1205733)[minusΠ

2+ Π

1+ Π

3]

120573

1015840

3=

1

4

119890

minus(1205731+1205732+1205733)[minusΠ

3+ Π

1+ Π

2]

120601

1015840= 120601

0119890

minus(1205731+1205732+1205733)

120590

1015840= 120590

0119890

minus(1205731+1205732+1205733)

(30)




Π1= Π

2+ 120575

2= Π

3+ 120575

3 (31)

with 1205752and 120575



by

1

Λ

Π

10158402

1minus 3Π

2

1+ ]Π

1+ 120575

1= 0 (32)


] = 2 (1205752+ 120575

3)

1205751(1199020) = (120575

2minus 120575

3)

2

minus 1199020

(33)


1199020=

16

11

Π

2

120601minus

6

11

Π120601Π120590+

1

33

Π

2

120590 (34)


Π1=

]6

+

1

6



1205731+ 120573

2+ 120573

3= ln(1

8

radic

]2 + 121205751

3Λ

sinh (radic3Λ120591)) (36)



120601 =

81206012

radic]2 + 121205751

ln(tanh(radic3Λ

2

120591))

120590 =

81205902

radic]2 + 121205751

ln(tanh(radic3Λ

2

120591))

(37)



Ω =

1

3

ln(18

radic

]2 + 121205751

3Λ


120573+=

2

3

(1205752minus 2120575

3)

radic]2 + 121205751

ln tanh(radic3Λ

2

120591)

120573minus= minus

2

radic3

1205752

radic]2 + 121205751

ln tanh(radic3Λ

2

120591)

(38)


120588119886=

Λ

3

(1205752minus 2120575

3)

2

+ 120575

2

2

]2 + 12 ((1205752minus 120575

3)

2

minus 1199020)

coth2 (radic3Λ

2

120591) (39)









1and 120590

1 respectively

Π

1015840

1= Π

1015840

2= Π

1015840

3

= 4Λ119890

1205731+1205732+1205733+ 16120587119866 (1 minus 120574) 120588

120574119890

minus120574(1205731+1205732+1205733)

120573

1015840

1=

1

4

119890

minus(1205731+1205732+1205733)[minusΠ

1+ Π

2+ Π

3]

120573

1015840

2=

1

4

119890

minus(1205731+1205732+1205733)[minusΠ

2+ Π

1+ Π

3]

120573

1015840

3=

1

4

119890

minus(1205731+1205732+1205733)[minusΠ

3+ Π

1+ Π

2]

120601

1015840= 120601

2119890

minus(1205731+1205732+1205733)

120590

1015840= 120590

2119890

minus(1205731+1205732+1205733)

(40)


1 120573

2 and 120573

3


Π1= Π

2+ 119896

1= Π

3+ 119896

2 (41)


120601 = 1206012int 119890

minus(1205731+1205732+1205733)119889120591

120590 = 1205902int 119890

minus(1205731+1205732+1205733)119889120591

(42)



120582 =

4Λ 120574 = 1

4Λ + 32120587119866120588minus1 120574 = minus1

(43)

for whichΠ10158401= 120582119890



4

120582

Π

10158402

1minus 3Π

2

1+ ]Π

1+ 120575

1= 0 (44)


Π1=

]6

+

1

6

radic]2 + 121205751cosh(

radic3120582

2

120591) (45)


1205731+ 120573

2+ 120573

3= ln(1

4

radic

1205753+ 12120575

1

3120582

sinh(radic3120582

2

120591)) (46)


120601 =

81206012

radic]2 + 121205751


4

120591)

120590 =

81205902

radic]2 + 121205751


4

120591)

(47)



Ω =

1

3

ln(14

radic

120575

2+ 12120575

1

3120582

sinh(radic3120582

2

120591))

120573+=

2

3

(1205752minus 2120575

3)

radic]2 + 121205751


2

120591)

+

1

6

ln(14

radic

]2 + 121205751

3120582

)

120573minus= minus

2

radic3

1205752

radic]2 + 121205751


2

120591)

minus

1

2radic3

ln(14

radic

]2 + 121205751

3120582

)

(48)


08)119905 and 120590 = (120590

08)119905



Π

1015840

1= Π

1015840

2= Π

1015840

3= 4Λ119890

1205731+1205732+1205733+ 120572

0 120572

0= 16120587119866120588

0

Π

1015840

120601= 0 Π

120601= 120601

1= 119888119905119890

(49)


1

Λ

Π

10158402

1minus 3Π

2

1+ 120583Π

1+ 120585

1= 0 (50)

where

1205851= 120575

1minus

120572

2

0

Λ

(51)


Π1=

]6

+

1

6



1205731+ 120573

2+ 120573

3= ln(1

8

radic

]2 + 121205851

3Λ

sinh (radic3Λ120591)) (53)


120601 =

81206012

radic]2 + 121205851

ln tanh(radic3Λ

2

120591)

120590 =

81205902

radic]2 + 121205851

ln tanh(radic3Λ

2

120591)

(54)


Ω =

1

3

ln(18

radic

]2 + 121205851

3Λ


120573+=

2

3

1205752minus 2120575

3

radic]2 + 121205851

ln tanh(radic3Λ

2

120591)

120573minus= minus

2

radic3

1205752

radic]2 + 121205851

ln tanh(radic3Λ

2

120591)

(55)




0 In this




5 Quantum Scheme



+ 120573

minus)




minus3ΩΠ

2

Ωas

minus119890

minus(3minus119876)Ω120597Ω119890

minus119876Ω120597Ω= minus119890

minus3Ω120597

2

Ω+ 119876119890

minus3Ω120597Ω (56)


◻Ψ + 119876

120597Ψ

120597Ω

minus

48

11

120597

2Ψ

1205971206012minus

1

11

120597

2Ψ

1205971205902+

18

11

120597

2Ψ

120597120601120597120590

minus [119887120574119890

3(1minus120574)Ω+ 48Λ119890

6Ω]Ψ = 0

(57)



+ 120573



Ψ = Φ (120601 120590) 120595 (Ω 120573plusmn) (58)


◻120595 + 119876

120597120595

120597Ω

minus [119887120574119890

3(1minus120574)Ω+ 48Λ119890

6Ωminus

120583

2

5

]120595 = 0 (59)

48

11

120597

2Φ

1205971206012+

1

11

120597

2Φ

1205971205902minus

18

11

120597

2Φ

120597120601120597120590

+ 120583

2Φ = 0 (60)


Φ(120601 120590) = 1198621sin (119862

3120601 + 119862

4120590 + 119862

5)

+ 1198622cos (119862

3120601 + 119862

4120590 + 119862

5)

(61)

where 119862119894

5



plusmn 120601 120590)Ψ =

119867119892(Ω 120573

plusmn)Ψ +

119867119898(120601 120590)Ψ = 0 where

119867119892y 119867



plusmn) = A(Ω)B

1(120573

+)B

2(120573



119889

2A

119889Ω2minus 119876

119889A

119889Ω

+ [119887120574119890

minus3(120574minus1)Ω+ 48Λ119890

6Ω+ 120590

2]A = 0

119889

2B1

1198891205732

+

+ 119886

2

2B

1= 0 997904rArr

B1= 120578

1119890

1198941198862120573++ 120578

2119890

minus1198941198862120573+

119889

2B2

1198891205732

minus

+ 119886

2

3B

2= 0 997904rArr

B2= 119888

1119890

1198941198863120573minus+ 119888

2119890

minus1198941198863120573minus

(62)

where 1205902 = 11988621+ 119886

2

2+ 119886

2

3and 119888

119894and 120578




119889

2A

119889Ω2minus 119876

119889A

119889Ω

+ [119887120574119890

minus3(1minus120574)Ω+ 120590

2]A = 0 (63)


119911 =

radic119887120574

119901

119890


(64)


119889A

119889Ω

=

119889A

119889119911

119889119911

119889Ω

= minus

3

2

(120574 minus 1) 119911

119889A

119889119911

119889

2A

119889Ω2=

9

4

(120574 minus 1)

2

119911

2 1198892A

1198891199112+

9

4

(120574 minus 1)

2

119911

119889A

119889119911

(65)


9

4

(120574 minus 1)

2

119911

2 1198892A

1198891199112+

9

4

(120574 minus 1)

2

119911

119889A

119889119911

minus

3

2

119876 (120574 minus 1) 119911

119889A

119889119911

+ [119901

2119911

2+ 120590

2] = 0

(66)


A = 119911

119902119876Φ (119911) (67)


9

4

(120574 minus 1)

2

119911

119902119876[119911

2 1198892Φ

1198891199112+ 119911(1 + 2119902119876

+

2

3

119876

120574 minus 1

)

119889Φ

119889119911

+ (

4119901

2

9 (120574 minus 1)

2119911

2

+ 119876

2119902

2+

2

3

119902

120574 minus 1

+

4120590

2

9 (120574 minus 1)

2)Φ] = 0

(68)


119911

2 1198892Φ

1198891199112+ 119911

119889Φ

119889119911

+ [119911

2minus

1

9 (120574 minus 1)

2(119876

2minus 4120590

2)]Φ

= 0

(69)


119902 = minus

1

3 (120574 minus 1)

119901 =

3

2

1003816100381610038161003816120574 minus 1

1003816100381610038161003816

(70)


119911 =

2radic119887120574

3

1003816100381610038161003816120574 minus 1

1003816100381610038161003816

119890


A = 119911

minus(1198763(120574minus1))Φ(119911)

(71)



A120574= 119888

120574(

2radic119887120574

3

1003816100381610038161003816120574 minus 1

1003816100381610038161003816

119890


minus1198763(120574minus1)

sdot 119885119894](

2radic119887120574

3

1003816100381610038161003816120574 minus 1

1003816100381610038161003816

119890


(72)

with

] = plusmnradic1

9 (120574 minus 1)

2(4120590

2minus 119876

2) (73)



int

infin

minusinfin

sech(120587120578

2

) 119869119894120578(119911) 119889120578 = 2 sin (119911) (74)


Ψ (Ω 120573plusmn 120601 120590) = Φ (120601 120590)


3

1003816100381610038161003816120574 minus 1

1003816100381610038161003816

119890


sdot [1205781119890

1198941198862120573++ 120578

2119890

minus1198941198862120573+]

sdot [1198881119890

1198941198863120573minus+ 119888

2119890

minus1198941198863120573minus]

(75)



119889

2A1

119889Ω2minus 119876

119889A1

119889Ω

+ 120590

2

1A1= 0 120590

2

1= 119887

1+ 120590

2

(76)

whose solution is

A1= 119860

1119890

((119876+radic1198762+41205902

1)2)Ω

+ 1198602119890

((119876minusradic1198762+41205902

1)2)Ω

(77)


Ψ (Ω 120573plusmn 120601 120590) = Φ (120601 120590)

sdot [1198601119890

((119876+radic1198762+41205902

1)2)Ω

+ 1198602119890

((119876minusradic1198762+41205902

1)2)Ω

]

sdot [1205781119890

1198941198862120573++ 120578

2119890

minus1198941198862120573+] [119888

1119890

1198941198863120573minus+ 119888

2119890

minus1198941198863120573minus]

(78)


A = 119890

(1198762)Ω119885] (4radic

Λ

3

119890

3Ω) ] = plusmn

1

6

radic1198762+ 4120590

2

1 (79)


13


119889

2Aminus1

119889Ω2minus 119876

119889Aminus1

119889Ω

+ [119889minus1119890

6Ωminus 120590

2]A

minus1= 0

119889minus1= 48120583

minus1+ 48Λ

(80)

with the solution

Aminus1= (

radic119889minus1

3

119890

3Ω)

1198766

119885] (radic119889

minus1

3

119890

3Ω)

] = plusmn1

6

radic1198762+ 4120590

2

(81)




119889

2A0

119889Ω2+ (48Λ119890

6Ω+ 119887

0119890

3Ω+ 120590

2)A

0= 0 119887

0= 48120583

0(82)

with the solution

A0= 119890

minus3Ω2[119863

1119872


(

8119894119890

3ΩradicΛ

radic3

)


(

8119894119890

3ΩradicΛ

radic3

)]

(83)

where119872119896119901

And119882119896119901


tegration constants

6 Final Remarks







plusmn 120601 120590)Ψ =

119867119892(Ω 120573

plusmn)Ψ +

119867119898(120601 120590)Ψ = 0 where 119867

119892y 119867



plusmn) In order




Appendix



119866119872119873

= 119890

Φ2119866

119864

119872119873 (A1)


119878 =

1

21205812

10

int119889

10119883radicminus119866(119890

Φ2R + 4119866

119864119872119873nabla119872Φnabla

119873Φ)

+ int119889


119864119890

5Φ2Lmatt

(A2)


R = 119890

minusΦ2(R

119864minus

9

2

119866

119864119872119873nabla119872nabla119873Φ

minus

9

2

119866

119864119872119873nabla119872Φnabla

119873Φ)

(A3)


119878 =

1

21205812

10

int119889


119864(R

119864minus

9

2

119866

119864119872119873nabla119872nabla119873Φ

minus

1

2

119866

119864119872119873nabla119872Φnabla

119873Φ) + int119889


119864119890

5Φ2Lmatt

(A4)


det119866119872119873

=119866 = 119890

minus12120590 (A5)


R(10)

=R(4)

minus 42

120583]nabla120583120590nabla]120590 + 12

120583]nabla120583nabla]120590 (A6)

we obtain that

119878 =

1

21205812

10

int119889

4119909119889


minus6120590119890

minus2Φ[R

(4)

minus 42

120583]nabla120583120590nabla]120590 + 12

120583]nabla120583nabla]120590 + 4


(A7)


Φ = 120601 minus

1

2

ln (119881) (A8)


119878 =

1

21205812

10

int119889


minus2(120601+3120590)[R

(4)

minus 42

120583]nabla120583120590nabla]120590

+ 12

120583]nabla120583nabla]120590 + 4

120583]nabla120583120601nabla]120601]

(A9)





120583] = 119890

2Θ119892120583] (A10)


R(4)

= 119890

minus2Θ(R

(4)minus 6119892


120583]nabla120583Θnabla]Θ) (A11)


Θ = 120601 + 3120590 + ln(120581

2

10

1205812

4

) (A12)


119878 =

1

21205812

4

int119889



120583]nabla120583nabla]120590

minus 2119892



120583]nabla120583120601nabla]120590)

(A13)


int119889

10119883radicminus119866119890

minus2Φ120581

2

10119890

2Φ119879119872119873119866

119872119873

= int119889

4119909radicminus119892119890

2(Θ+120601)119879119872119873119866

119872119873

= int119889

4119909radicminus119892 (119890

2120601119879120583]119892

120583]+ 119890

2(Θ+120601)119879119898119899

119898119899)

(A14)


Competing Interests


Acknowledgments


References


























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







119879119872119873

= (

119879120583] 0

0119879119898119899

) (6)

where 119879120583] and 119879


components of 119879119872119873





119879120583] = 119890

2120601119879120583] (7)





119889119904

2= minus119873 (119905) 119889119905

2+ 119890

2Ω(119905)(119890

2120573(119905))

119894119895120596

119894120596

119895 (8)


matrix 120573119894119895= diag(120573

++radic3120573

minus 120573

+minusradic3120573

minus minus2120573

+)Ω(119905) and 120573

plusmn


119894= (12)119862

119894

119895119896120596

119895and 120596

119896 with119862

119894



119889119904

2

119868= minus119873

2119889119905

2+ 119877

2

1119889119909

2+ 119877

2

2119889119910

2+ 119877

2

3119889119911

2 (9)

where 119877119894

3


1198771= 119890

Ω+120573++radic3120573

minus

1198772= 119890


minus

1198773= 119890

Ωminus2120573+

119881 = 119877111987721198773= 119890

3Ω

(10)



119879120583] = (119901 + 120588) 119906120583119906] + 119901119892120583] (11)






= 16119873120587119866119873119862120574119890

minus3(1+120574)Ω+ 2119873Λ119890

3Ω

(12)


L119868=

119890

3Ω

119873

[6Ω

2

minus 6120573

2

+minus 6

120573

2

minus+ 96

2+ 36

120601 + 2

120601

2

+ 16120587119866119873

2120588 + 2Λ119873

2]

(13)


120597L120597120583 where 119902120583 = (Ω 120573+ 120573


ΠΩ=

12119890

3ΩΩ

119873

Ω =

119873119890

minus3ΩΠΩ

12

Πplusmn= minus

12119890

3Ω120573plusmn

119873

120573plusmn= minus

119873119890


12

Π120601=

119890

3Ω

119873

[36 + 4120601]

120601 =

119873119890

minus3Ω

44

[3Π120590minus 16Π

120601]

Π120590=

119890

3Ω

119873

[192 + 36120601] =

119873119890

minus3Ω

132

[9Π120601minus Π

120590]

(14)







H119868=

119890

minus3Ω

24

[Π

2

Ωminus Π

2

+minus Π

2

minusminus

48

11

Π

2

120601+

18

11

Π120601Π120590

minus

1

11

Π

2

120590minus 384120587119866

119873120588120574119890

3(1minus120574)Ωminus 48Λ119890

6Ω]

(15)




1205731= Ω + 120573

++radic3120573

minus

1205732= Ω + 120573

+minusradic3120573

minus

1205733= Ω minus 2120573

+

(16)


L119868=

119890

1205731+1205732+1205733

119873

(2120601

2

+ 36120601 + 96

2+ 2

1205731

1205732

+ 21205731

1205733+ 2

1205733

1205732+ 2Λ119873

2

+ 16120587119866119873

2120588120574119890

minus(1+120574)(1205731+1205732+1205733))

(17)


H119868=

1

8

119890

minus(1205731+1205732+1205733)[minusΠ

2

1minus Π

2

2minus Π

2

3+ 2Π

1Π2

+ 2Π1Π3+ 2Π

2Π3minus

16

11

Π

2

120601+

6

11

Π120601Π120590minus

1

33

Π

2

120590

minus 128120587119866120588120574119890

(1minus120574)(1205731+1205732+1205733)minus 16Λ119890

2(1205731+1205732+1205733)]

(18)



Ω

10158402= 120573

10158402

++ 120573

10158402

minus+ 16120590

10158402+

1

3

120601

10158402+ 6120601

1015840120590

1015840

+

8

3

120587119866120583120574119890

minus3(1+120574)Ω+

Λ

3

(19)




++120573

10158402

minus)119867

2rarr 0

(120573

10158402

++120573

10158402




120588119886= 120573

10158402

++ 120573

10158402

minus (20)




120601= 119901

120601= constant

Π120590= 0 997888rarr Π

120590= 119901

120590= constant

Π+= 0 997888rarr Π

+= 119901

+= constant


minus= 119901

minus= constant

(21)


Ωas 1205812

Ωsim 119901

2

++119901

2

minus+(16132

2)(119901

120590minus

9119901120601)

2+(1(3sdot44

2))(3119901

120590minus16119901

120601)

2+(1(22sdot44))(119901

120590minus9119901

120601)(3119901

120590minus

16119901120601) we have that

120588119886sim 119890

minus6Ω

120588120601120590sim 119890

minus6Ω

Ω

10158402sim

Λ

3

+ 120581Ω

2119890

minus6Ω+ 119887

120574119890

minus3(1+120574)Ω

119887120574=

8

3

120587119866119873120583120574

(22)


120588120601120590

sim constant

120588119886

120588120574

sim 119890

3Ω(120574minus1)

120588119886

Ω10158402sim

1

120581Ω

2+ (Λ3) 119890

6Ω+ 119887

1205741198903(1minus120574)Ω

(23)


4 Case of Interest





that

H119868vac=

1

8

119890

minus(1205731+1205732+1205733)[minusΠ

2

1minus Π

2

2minus Π

2

3+ 2Π

1Π2

+ 2Π1Π3+ 2Π

2Π3minus

16

11

Π

2

120601+

6

11

Π120601Π120590minus

1

33

Π

2

120590]

(24)



119875119894= 120597H120597119902

119894

become


119894= constant

Π120601= 0 997904rArr Π

120601= constant

Π120590= 0 997904rArr Π

120590= constant

120573

1015840

1=

1

4

119890

minus(1205731+1205732+1205733)[minusΠ

1+ Π

2+ Π

3]

120573

1015840

2=

1

4

119890

minus(1205731+1205732+1205733)[minusΠ

2+ Π

1+ Π

3]

120573

1015840

3=

1

4

119890

minus(1205731+1205732+1205733)[minusΠ

3+ Π

1+ Π

2]

120601

1015840= 120601

0119890

minus(1205731+1205732+1205733)

120590

1015840= 120590

0119890

minus(1205731+1205732+1205733)

(25)



1205731+ 120573

2+ 120573

3= ln [120598120591 + 119887

0] 120598 = 119887

1+ 119887

2+ 119887

3 (26)

with 119887119894= Π

119894and 119887



Ω = ln (120598120591 + 1198870)

13

120573+= ln (120598120591 + 119887

0)

(120598minus31198873)6120598

120573minus= ln (120598120591 + 119887

0)

(1198873+21198871minus120598)2radic3120598

(27)


120601 =

1206010

120598

ln (120598120591 + 1198870)

120590 =

1205900

120598

ln (120598120591 + 1198870)

(28)

where 1206010= Π

120601and 120590

0= Π




0


H119868=

1

8

119890

minus(1205731+1205732+1205733)[minusΠ

2

1minus Π

2

2minus Π

2

3+ 2Π

1Π2

+ 2Π1Π3+ 2Π

2Π3minus

16

11

Π

2

120601+

6

11

Π120601Π120590minus

1

33

Π

2

120590

minus 16Λ119890

2(1205731+1205732+1205733)]

(29)


Π

1015840

1= Π

1015840

2= Π

1015840

3= 4Λ119890

1205731+1205732+1205733

120573

1015840

1=

1

4

119890

minus(1205731+1205732+1205733)[minusΠ

1+ Π

2+ Π

3]

120573

1015840

2=

1

4

119890

minus(1205731+1205732+1205733)[minusΠ

2+ Π

1+ Π

3]

120573

1015840

3=

1

4

119890

minus(1205731+1205732+1205733)[minusΠ

3+ Π

1+ Π

2]

120601

1015840= 120601

0119890

minus(1205731+1205732+1205733)

120590

1015840= 120590

0119890

minus(1205731+1205732+1205733)

(30)




Π1= Π

2+ 120575

2= Π

3+ 120575

3 (31)

with 1205752and 120575



by

1

Λ

Π

10158402

1minus 3Π

2

1+ ]Π

1+ 120575

1= 0 (32)


] = 2 (1205752+ 120575

3)

1205751(1199020) = (120575

2minus 120575

3)

2

minus 1199020

(33)


1199020=

16

11

Π

2

120601minus

6

11

Π120601Π120590+

1

33

Π

2

120590 (34)


Π1=

]6

+

1

6



1205731+ 120573

2+ 120573

3= ln(1

8

radic

]2 + 121205751

3Λ

sinh (radic3Λ120591)) (36)



120601 =

81206012

radic]2 + 121205751

ln(tanh(radic3Λ

2

120591))

120590 =

81205902

radic]2 + 121205751

ln(tanh(radic3Λ

2

120591))

(37)



Ω =

1

3

ln(18

radic

]2 + 121205751

3Λ


120573+=

2

3

(1205752minus 2120575

3)

radic]2 + 121205751

ln tanh(radic3Λ

2

120591)

120573minus= minus

2

radic3

1205752

radic]2 + 121205751

ln tanh(radic3Λ

2

120591)

(38)


120588119886=

Λ

3

(1205752minus 2120575

3)

2

+ 120575

2

2

]2 + 12 ((1205752minus 120575

3)

2

minus 1199020)

coth2 (radic3Λ

2

120591) (39)









1and 120590

1 respectively

Π

1015840

1= Π

1015840

2= Π

1015840

3

= 4Λ119890

1205731+1205732+1205733+ 16120587119866 (1 minus 120574) 120588

120574119890

minus120574(1205731+1205732+1205733)

120573

1015840

1=

1

4

119890

minus(1205731+1205732+1205733)[minusΠ

1+ Π

2+ Π

3]

120573

1015840

2=

1

4

119890

minus(1205731+1205732+1205733)[minusΠ

2+ Π

1+ Π

3]

120573

1015840

3=

1

4

119890

minus(1205731+1205732+1205733)[minusΠ

3+ Π

1+ Π

2]

120601

1015840= 120601

2119890

minus(1205731+1205732+1205733)

120590

1015840= 120590

2119890

minus(1205731+1205732+1205733)

(40)


1 120573

2 and 120573

3


Π1= Π

2+ 119896

1= Π

3+ 119896

2 (41)


120601 = 1206012int 119890

minus(1205731+1205732+1205733)119889120591

120590 = 1205902int 119890

minus(1205731+1205732+1205733)119889120591

(42)



120582 =

4Λ 120574 = 1

4Λ + 32120587119866120588minus1 120574 = minus1

(43)

for whichΠ10158401= 120582119890



4

120582

Π

10158402

1minus 3Π

2

1+ ]Π

1+ 120575

1= 0 (44)


Π1=

]6

+

1

6

radic]2 + 121205751cosh(

radic3120582

2

120591) (45)


1205731+ 120573

2+ 120573

3= ln(1

4

radic

1205753+ 12120575

1

3120582

sinh(radic3120582

2

120591)) (46)


120601 =

81206012

radic]2 + 121205751


4

120591)

120590 =

81205902

radic]2 + 121205751


4

120591)

(47)



Ω =

1

3

ln(14

radic

120575

2+ 12120575

1

3120582

sinh(radic3120582

2

120591))

120573+=

2

3

(1205752minus 2120575

3)

radic]2 + 121205751


2

120591)

+

1

6

ln(14

radic

]2 + 121205751

3120582

)

120573minus= minus

2

radic3

1205752

radic]2 + 121205751


2

120591)

minus

1

2radic3

ln(14

radic

]2 + 121205751

3120582

)

(48)


08)119905 and 120590 = (120590

08)119905



Π

1015840

1= Π

1015840

2= Π

1015840

3= 4Λ119890

1205731+1205732+1205733+ 120572

0 120572

0= 16120587119866120588

0

Π

1015840

120601= 0 Π

120601= 120601

1= 119888119905119890

(49)


1

Λ

Π

10158402

1minus 3Π

2

1+ 120583Π

1+ 120585

1= 0 (50)

where

1205851= 120575

1minus

120572

2

0

Λ

(51)


Π1=

]6

+

1

6



1205731+ 120573

2+ 120573

3= ln(1

8

radic

]2 + 121205851

3Λ

sinh (radic3Λ120591)) (53)


120601 =

81206012

radic]2 + 121205851

ln tanh(radic3Λ

2

120591)

120590 =

81205902

radic]2 + 121205851

ln tanh(radic3Λ

2

120591)

(54)


Ω =

1

3

ln(18

radic

]2 + 121205851

3Λ


120573+=

2

3

1205752minus 2120575

3

radic]2 + 121205851

ln tanh(radic3Λ

2

120591)

120573minus= minus

2

radic3

1205752

radic]2 + 121205851

ln tanh(radic3Λ

2

120591)

(55)




0 In this




5 Quantum Scheme



+ 120573

minus)




minus3ΩΠ

2

Ωas

minus119890

minus(3minus119876)Ω120597Ω119890

minus119876Ω120597Ω= minus119890

minus3Ω120597

2

Ω+ 119876119890

minus3Ω120597Ω (56)


◻Ψ + 119876

120597Ψ

120597Ω

minus

48

11

120597

2Ψ

1205971206012minus

1

11

120597

2Ψ

1205971205902+

18

11

120597

2Ψ

120597120601120597120590

minus [119887120574119890

3(1minus120574)Ω+ 48Λ119890

6Ω]Ψ = 0

(57)



+ 120573



Ψ = Φ (120601 120590) 120595 (Ω 120573plusmn) (58)


◻120595 + 119876

120597120595

120597Ω

minus [119887120574119890

3(1minus120574)Ω+ 48Λ119890

6Ωminus

120583

2

5

]120595 = 0 (59)

48

11

120597

2Φ

1205971206012+

1

11

120597

2Φ

1205971205902minus

18

11

120597

2Φ

120597120601120597120590

+ 120583

2Φ = 0 (60)


Φ(120601 120590) = 1198621sin (119862

3120601 + 119862

4120590 + 119862

5)

+ 1198622cos (119862

3120601 + 119862

4120590 + 119862

5)

(61)

where 119862119894

5



plusmn 120601 120590)Ψ =

119867119892(Ω 120573

plusmn)Ψ +

119867119898(120601 120590)Ψ = 0 where

119867119892y 119867



plusmn) = A(Ω)B

1(120573

+)B

2(120573



119889

2A

119889Ω2minus 119876

119889A

119889Ω

+ [119887120574119890

minus3(120574minus1)Ω+ 48Λ119890

6Ω+ 120590

2]A = 0

119889

2B1

1198891205732

+

+ 119886

2

2B

1= 0 997904rArr

B1= 120578

1119890

1198941198862120573++ 120578

2119890

minus1198941198862120573+

119889

2B2

1198891205732

minus

+ 119886

2

3B

2= 0 997904rArr

B2= 119888

1119890

1198941198863120573minus+ 119888

2119890

minus1198941198863120573minus

(62)

where 1205902 = 11988621+ 119886

2

2+ 119886

2

3and 119888

119894and 120578




119889

2A

119889Ω2minus 119876

119889A

119889Ω

+ [119887120574119890

minus3(1minus120574)Ω+ 120590

2]A = 0 (63)


119911 =

radic119887120574

119901

119890


(64)


119889A

119889Ω

=

119889A

119889119911

119889119911

119889Ω

= minus

3

2

(120574 minus 1) 119911

119889A

119889119911

119889

2A

119889Ω2=

9

4

(120574 minus 1)

2

119911

2 1198892A

1198891199112+

9

4

(120574 minus 1)

2

119911

119889A

119889119911

(65)


9

4

(120574 minus 1)

2

119911

2 1198892A

1198891199112+

9

4

(120574 minus 1)

2

119911

119889A

119889119911

minus

3

2

119876 (120574 minus 1) 119911

119889A

119889119911

+ [119901

2119911

2+ 120590

2] = 0

(66)


A = 119911

119902119876Φ (119911) (67)


9

4

(120574 minus 1)

2

119911

119902119876[119911

2 1198892Φ

1198891199112+ 119911(1 + 2119902119876

+

2

3

119876

120574 minus 1

)

119889Φ

119889119911

+ (

4119901

2

9 (120574 minus 1)

2119911

2

+ 119876

2119902

2+

2

3

119902

120574 minus 1

+

4120590

2

9 (120574 minus 1)

2)Φ] = 0

(68)


119911

2 1198892Φ

1198891199112+ 119911

119889Φ

119889119911

+ [119911

2minus

1

9 (120574 minus 1)

2(119876

2minus 4120590

2)]Φ

= 0

(69)


119902 = minus

1

3 (120574 minus 1)

119901 =

3

2

1003816100381610038161003816120574 minus 1

1003816100381610038161003816

(70)


119911 =

2radic119887120574

3

1003816100381610038161003816120574 minus 1

1003816100381610038161003816

119890


A = 119911

minus(1198763(120574minus1))Φ(119911)

(71)



A120574= 119888

120574(

2radic119887120574

3

1003816100381610038161003816120574 minus 1

1003816100381610038161003816

119890


minus1198763(120574minus1)

sdot 119885119894](

2radic119887120574

3

1003816100381610038161003816120574 minus 1

1003816100381610038161003816

119890


(72)

with

] = plusmnradic1

9 (120574 minus 1)

2(4120590

2minus 119876

2) (73)



int

infin

minusinfin

sech(120587120578

2

) 119869119894120578(119911) 119889120578 = 2 sin (119911) (74)


Ψ (Ω 120573plusmn 120601 120590) = Φ (120601 120590)


3

1003816100381610038161003816120574 minus 1

1003816100381610038161003816

119890


sdot [1205781119890

1198941198862120573++ 120578

2119890

minus1198941198862120573+]

sdot [1198881119890

1198941198863120573minus+ 119888

2119890

minus1198941198863120573minus]

(75)



119889

2A1

119889Ω2minus 119876

119889A1

119889Ω

+ 120590

2

1A1= 0 120590

2

1= 119887

1+ 120590

2

(76)

whose solution is

A1= 119860

1119890

((119876+radic1198762+41205902

1)2)Ω

+ 1198602119890

((119876minusradic1198762+41205902

1)2)Ω

(77)


Ψ (Ω 120573plusmn 120601 120590) = Φ (120601 120590)

sdot [1198601119890

((119876+radic1198762+41205902

1)2)Ω

+ 1198602119890

((119876minusradic1198762+41205902

1)2)Ω

]

sdot [1205781119890

1198941198862120573++ 120578

2119890

minus1198941198862120573+] [119888

1119890

1198941198863120573minus+ 119888

2119890

minus1198941198863120573minus]

(78)


A = 119890

(1198762)Ω119885] (4radic

Λ

3

119890

3Ω) ] = plusmn

1

6

radic1198762+ 4120590

2

1 (79)


13


119889

2Aminus1

119889Ω2minus 119876

119889Aminus1

119889Ω

+ [119889minus1119890

6Ωminus 120590

2]A

minus1= 0

119889minus1= 48120583

minus1+ 48Λ

(80)

with the solution

Aminus1= (

radic119889minus1

3

119890

3Ω)

1198766

119885] (radic119889

minus1

3

119890

3Ω)

] = plusmn1

6

radic1198762+ 4120590

2

(81)




119889

2A0

119889Ω2+ (48Λ119890

6Ω+ 119887

0119890

3Ω+ 120590

2)A

0= 0 119887

0= 48120583

0(82)

with the solution

A0= 119890

minus3Ω2[119863

1119872


(

8119894119890

3ΩradicΛ

radic3

)


(

8119894119890

3ΩradicΛ

radic3

)]

(83)

where119872119896119901

And119882119896119901


tegration constants

6 Final Remarks







plusmn 120601 120590)Ψ =

119867119892(Ω 120573

plusmn)Ψ +

119867119898(120601 120590)Ψ = 0 where 119867

119892y 119867



plusmn) In order




Appendix



119866119872119873

= 119890

Φ2119866

119864

119872119873 (A1)


119878 =

1

21205812

10

int119889

10119883radicminus119866(119890

Φ2R + 4119866

119864119872119873nabla119872Φnabla

119873Φ)

+ int119889


119864119890

5Φ2Lmatt

(A2)


R = 119890

minusΦ2(R

119864minus

9

2

119866

119864119872119873nabla119872nabla119873Φ

minus

9

2

119866

119864119872119873nabla119872Φnabla

119873Φ)

(A3)


119878 =

1

21205812

10

int119889


119864(R

119864minus

9

2

119866

119864119872119873nabla119872nabla119873Φ

minus

1

2

119866

119864119872119873nabla119872Φnabla

119873Φ) + int119889


119864119890

5Φ2Lmatt

(A4)


det119866119872119873

=119866 = 119890

minus12120590 (A5)


R(10)

=R(4)

minus 42

120583]nabla120583120590nabla]120590 + 12

120583]nabla120583nabla]120590 (A6)

we obtain that

119878 =

1

21205812

10

int119889

4119909119889


minus6120590119890

minus2Φ[R

(4)

minus 42

120583]nabla120583120590nabla]120590 + 12

120583]nabla120583nabla]120590 + 4


(A7)


Φ = 120601 minus

1

2

ln (119881) (A8)


119878 =

1

21205812

10

int119889


minus2(120601+3120590)[R

(4)

minus 42

120583]nabla120583120590nabla]120590

+ 12

120583]nabla120583nabla]120590 + 4

120583]nabla120583120601nabla]120601]

(A9)





120583] = 119890

2Θ119892120583] (A10)


R(4)

= 119890

minus2Θ(R

(4)minus 6119892


120583]nabla120583Θnabla]Θ) (A11)


Θ = 120601 + 3120590 + ln(120581

2

10

1205812

4

) (A12)


119878 =

1

21205812

4

int119889



120583]nabla120583nabla]120590

minus 2119892



120583]nabla120583120601nabla]120590)

(A13)


int119889

10119883radicminus119866119890

minus2Φ120581

2

10119890

2Φ119879119872119873119866

119872119873

= int119889

4119909radicminus119892119890

2(Θ+120601)119879119872119873119866

119872119873

= int119889

4119909radicminus119892 (119890

2120601119879120583]119892

120583]+ 119890

2(Θ+120601)119879119898119899

119898119899)

(A14)


Competing Interests


Acknowledgments


References


























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





H119868=

119890

minus3Ω

24

[Π

2

Ωminus Π

2

+minus Π

2

minusminus

48

11

Π

2

120601+

18

11

Π120601Π120590

minus

1

11

Π

2

120590minus 384120587119866

119873120588120574119890

3(1minus120574)Ωminus 48Λ119890

6Ω]

(15)




1205731= Ω + 120573

++radic3120573

minus

1205732= Ω + 120573

+minusradic3120573

minus

1205733= Ω minus 2120573

+

(16)


L119868=

119890

1205731+1205732+1205733

119873

(2120601

2

+ 36120601 + 96

2+ 2

1205731

1205732

+ 21205731

1205733+ 2

1205733

1205732+ 2Λ119873

2

+ 16120587119866119873

2120588120574119890

minus(1+120574)(1205731+1205732+1205733))

(17)


H119868=

1

8

119890

minus(1205731+1205732+1205733)[minusΠ

2

1minus Π

2

2minus Π

2

3+ 2Π

1Π2

+ 2Π1Π3+ 2Π

2Π3minus

16

11

Π

2

120601+

6

11

Π120601Π120590minus

1

33

Π

2

120590

minus 128120587119866120588120574119890

(1minus120574)(1205731+1205732+1205733)minus 16Λ119890

2(1205731+1205732+1205733)]

(18)



Ω

10158402= 120573

10158402

++ 120573

10158402

minus+ 16120590

10158402+

1

3

120601

10158402+ 6120601

1015840120590

1015840

+

8

3

120587119866120583120574119890

minus3(1+120574)Ω+

Λ

3

(19)




++120573

10158402

minus)119867

2rarr 0

(120573

10158402

++120573

10158402




120588119886= 120573

10158402

++ 120573

10158402

minus (20)




120601= 119901

120601= constant

Π120590= 0 997888rarr Π

120590= 119901

120590= constant

Π+= 0 997888rarr Π

+= 119901

+= constant


minus= 119901

minus= constant

(21)


Ωas 1205812

Ωsim 119901

2

++119901

2

minus+(16132

2)(119901

120590minus

9119901120601)

2+(1(3sdot44

2))(3119901

120590minus16119901

120601)

2+(1(22sdot44))(119901

120590minus9119901

120601)(3119901

120590minus

16119901120601) we have that

120588119886sim 119890

minus6Ω

120588120601120590sim 119890

minus6Ω

Ω

10158402sim

Λ

3

+ 120581Ω

2119890

minus6Ω+ 119887

120574119890

minus3(1+120574)Ω

119887120574=

8

3

120587119866119873120583120574

(22)


120588120601120590

sim constant

120588119886

120588120574

sim 119890

3Ω(120574minus1)

120588119886

Ω10158402sim

1

120581Ω

2+ (Λ3) 119890

6Ω+ 119887

1205741198903(1minus120574)Ω

(23)


4 Case of Interest





that

H119868vac=

1

8

119890

minus(1205731+1205732+1205733)[minusΠ

2

1minus Π

2

2minus Π

2

3+ 2Π

1Π2

+ 2Π1Π3+ 2Π

2Π3minus

16

11

Π

2

120601+

6

11

Π120601Π120590minus

1

33

Π

2

120590]

(24)



119875119894= 120597H120597119902

119894

become


119894= constant

Π120601= 0 997904rArr Π

120601= constant

Π120590= 0 997904rArr Π

120590= constant

120573

1015840

1=

1

4

119890

minus(1205731+1205732+1205733)[minusΠ

1+ Π

2+ Π

3]

120573

1015840

2=

1

4

119890

minus(1205731+1205732+1205733)[minusΠ

2+ Π

1+ Π

3]

120573

1015840

3=

1

4

119890

minus(1205731+1205732+1205733)[minusΠ

3+ Π

1+ Π

2]

120601

1015840= 120601

0119890

minus(1205731+1205732+1205733)

120590

1015840= 120590

0119890

minus(1205731+1205732+1205733)

(25)



1205731+ 120573

2+ 120573

3= ln [120598120591 + 119887

0] 120598 = 119887

1+ 119887

2+ 119887

3 (26)

with 119887119894= Π

119894and 119887



Ω = ln (120598120591 + 1198870)

13

120573+= ln (120598120591 + 119887

0)

(120598minus31198873)6120598

120573minus= ln (120598120591 + 119887

0)

(1198873+21198871minus120598)2radic3120598

(27)


120601 =

1206010

120598

ln (120598120591 + 1198870)

120590 =

1205900

120598

ln (120598120591 + 1198870)

(28)

where 1206010= Π

120601and 120590

0= Π




0


H119868=

1

8

119890

minus(1205731+1205732+1205733)[minusΠ

2

1minus Π

2

2minus Π

2

3+ 2Π

1Π2

+ 2Π1Π3+ 2Π

2Π3minus

16

11

Π

2

120601+

6

11

Π120601Π120590minus

1

33

Π

2

120590

minus 16Λ119890

2(1205731+1205732+1205733)]

(29)


Π

1015840

1= Π

1015840

2= Π

1015840

3= 4Λ119890

1205731+1205732+1205733

120573

1015840

1=

1

4

119890

minus(1205731+1205732+1205733)[minusΠ

1+ Π

2+ Π

3]

120573

1015840

2=

1

4

119890

minus(1205731+1205732+1205733)[minusΠ

2+ Π

1+ Π

3]

120573

1015840

3=

1

4

119890

minus(1205731+1205732+1205733)[minusΠ

3+ Π

1+ Π

2]

120601

1015840= 120601

0119890

minus(1205731+1205732+1205733)

120590

1015840= 120590

0119890

minus(1205731+1205732+1205733)

(30)




Π1= Π

2+ 120575

2= Π

3+ 120575

3 (31)

with 1205752and 120575



by

1

Λ

Π

10158402

1minus 3Π

2

1+ ]Π

1+ 120575

1= 0 (32)


] = 2 (1205752+ 120575

3)

1205751(1199020) = (120575

2minus 120575

3)

2

minus 1199020

(33)


1199020=

16

11

Π

2

120601minus

6

11

Π120601Π120590+

1

33

Π

2

120590 (34)


Π1=

]6

+

1

6



1205731+ 120573

2+ 120573

3= ln(1

8

radic

]2 + 121205751

3Λ

sinh (radic3Λ120591)) (36)



120601 =

81206012

radic]2 + 121205751

ln(tanh(radic3Λ

2

120591))

120590 =

81205902

radic]2 + 121205751

ln(tanh(radic3Λ

2

120591))

(37)



Ω =

1

3

ln(18

radic

]2 + 121205751

3Λ


120573+=

2

3

(1205752minus 2120575

3)

radic]2 + 121205751

ln tanh(radic3Λ

2

120591)

120573minus= minus

2

radic3

1205752

radic]2 + 121205751

ln tanh(radic3Λ

2

120591)

(38)


120588119886=

Λ

3

(1205752minus 2120575

3)

2

+ 120575

2

2

]2 + 12 ((1205752minus 120575

3)

2

minus 1199020)

coth2 (radic3Λ

2

120591) (39)









1and 120590

1 respectively

Π

1015840

1= Π

1015840

2= Π

1015840

3

= 4Λ119890

1205731+1205732+1205733+ 16120587119866 (1 minus 120574) 120588

120574119890

minus120574(1205731+1205732+1205733)

120573

1015840

1=

1

4

119890

minus(1205731+1205732+1205733)[minusΠ

1+ Π

2+ Π

3]

120573

1015840

2=

1

4

119890

minus(1205731+1205732+1205733)[minusΠ

2+ Π

1+ Π

3]

120573

1015840

3=

1

4

119890

minus(1205731+1205732+1205733)[minusΠ

3+ Π

1+ Π

2]

120601

1015840= 120601

2119890

minus(1205731+1205732+1205733)

120590

1015840= 120590

2119890

minus(1205731+1205732+1205733)

(40)


1 120573

2 and 120573

3


Π1= Π

2+ 119896

1= Π

3+ 119896

2 (41)


120601 = 1206012int 119890

minus(1205731+1205732+1205733)119889120591

120590 = 1205902int 119890

minus(1205731+1205732+1205733)119889120591

(42)



120582 =

4Λ 120574 = 1

4Λ + 32120587119866120588minus1 120574 = minus1

(43)

for whichΠ10158401= 120582119890



4

120582

Π

10158402

1minus 3Π

2

1+ ]Π

1+ 120575

1= 0 (44)


Π1=

]6

+

1

6

radic]2 + 121205751cosh(

radic3120582

2

120591) (45)


1205731+ 120573

2+ 120573

3= ln(1

4

radic

1205753+ 12120575

1

3120582

sinh(radic3120582

2

120591)) (46)


120601 =

81206012

radic]2 + 121205751


4

120591)

120590 =

81205902

radic]2 + 121205751


4

120591)

(47)



Ω =

1

3

ln(14

radic

120575

2+ 12120575

1

3120582

sinh(radic3120582

2

120591))

120573+=

2

3

(1205752minus 2120575

3)

radic]2 + 121205751


2

120591)

+

1

6

ln(14

radic

]2 + 121205751

3120582

)

120573minus= minus

2

radic3

1205752

radic]2 + 121205751


2

120591)

minus

1

2radic3

ln(14

radic

]2 + 121205751

3120582

)

(48)


08)119905 and 120590 = (120590

08)119905



Π

1015840

1= Π

1015840

2= Π

1015840

3= 4Λ119890

1205731+1205732+1205733+ 120572

0 120572

0= 16120587119866120588

0

Π

1015840

120601= 0 Π

120601= 120601

1= 119888119905119890

(49)


1

Λ

Π

10158402

1minus 3Π

2

1+ 120583Π

1+ 120585

1= 0 (50)

where

1205851= 120575

1minus

120572

2

0

Λ

(51)


Π1=

]6

+

1

6



1205731+ 120573

2+ 120573

3= ln(1

8

radic

]2 + 121205851

3Λ

sinh (radic3Λ120591)) (53)


120601 =

81206012

radic]2 + 121205851

ln tanh(radic3Λ

2

120591)

120590 =

81205902

radic]2 + 121205851

ln tanh(radic3Λ

2

120591)

(54)


Ω =

1

3

ln(18

radic

]2 + 121205851

3Λ


120573+=

2

3

1205752minus 2120575

3

radic]2 + 121205851

ln tanh(radic3Λ

2

120591)

120573minus= minus

2

radic3

1205752

radic]2 + 121205851

ln tanh(radic3Λ

2

120591)

(55)




0 In this




5 Quantum Scheme



+ 120573

minus)




minus3ΩΠ

2

Ωas

minus119890

minus(3minus119876)Ω120597Ω119890

minus119876Ω120597Ω= minus119890

minus3Ω120597

2

Ω+ 119876119890

minus3Ω120597Ω (56)


◻Ψ + 119876

120597Ψ

120597Ω

minus

48

11

120597

2Ψ

1205971206012minus

1

11

120597

2Ψ

1205971205902+

18

11

120597

2Ψ

120597120601120597120590

minus [119887120574119890

3(1minus120574)Ω+ 48Λ119890

6Ω]Ψ = 0

(57)



+ 120573



Ψ = Φ (120601 120590) 120595 (Ω 120573plusmn) (58)


◻120595 + 119876

120597120595

120597Ω

minus [119887120574119890

3(1minus120574)Ω+ 48Λ119890

6Ωminus

120583

2

5

]120595 = 0 (59)

48

11

120597

2Φ

1205971206012+

1

11

120597

2Φ

1205971205902minus

18

11

120597

2Φ

120597120601120597120590

+ 120583

2Φ = 0 (60)


Φ(120601 120590) = 1198621sin (119862

3120601 + 119862

4120590 + 119862

5)

+ 1198622cos (119862

3120601 + 119862

4120590 + 119862

5)

(61)

where 119862119894

5



plusmn 120601 120590)Ψ =

119867119892(Ω 120573

plusmn)Ψ +

119867119898(120601 120590)Ψ = 0 where

119867119892y 119867



plusmn) = A(Ω)B

1(120573

+)B

2(120573



119889

2A

119889Ω2minus 119876

119889A

119889Ω

+ [119887120574119890

minus3(120574minus1)Ω+ 48Λ119890

6Ω+ 120590

2]A = 0

119889

2B1

1198891205732

+

+ 119886

2

2B

1= 0 997904rArr

B1= 120578

1119890

1198941198862120573++ 120578

2119890

minus1198941198862120573+

119889

2B2

1198891205732

minus

+ 119886

2

3B

2= 0 997904rArr

B2= 119888

1119890

1198941198863120573minus+ 119888

2119890

minus1198941198863120573minus

(62)

where 1205902 = 11988621+ 119886

2

2+ 119886

2

3and 119888

119894and 120578




119889

2A

119889Ω2minus 119876

119889A

119889Ω

+ [119887120574119890

minus3(1minus120574)Ω+ 120590

2]A = 0 (63)


119911 =

radic119887120574

119901

119890


(64)


119889A

119889Ω

=

119889A

119889119911

119889119911

119889Ω

= minus

3

2

(120574 minus 1) 119911

119889A

119889119911

119889

2A

119889Ω2=

9

4

(120574 minus 1)

2

119911

2 1198892A

1198891199112+

9

4

(120574 minus 1)

2

119911

119889A

119889119911

(65)


9

4

(120574 minus 1)

2

119911

2 1198892A

1198891199112+

9

4

(120574 minus 1)

2

119911

119889A

119889119911

minus

3

2

119876 (120574 minus 1) 119911

119889A

119889119911

+ [119901

2119911

2+ 120590

2] = 0

(66)


A = 119911

119902119876Φ (119911) (67)


9

4

(120574 minus 1)

2

119911

119902119876[119911

2 1198892Φ

1198891199112+ 119911(1 + 2119902119876

+

2

3

119876

120574 minus 1

)

119889Φ

119889119911

+ (

4119901

2

9 (120574 minus 1)

2119911

2

+ 119876

2119902

2+

2

3

119902

120574 minus 1

+

4120590

2

9 (120574 minus 1)

2)Φ] = 0

(68)


119911

2 1198892Φ

1198891199112+ 119911

119889Φ

119889119911

+ [119911

2minus

1

9 (120574 minus 1)

2(119876

2minus 4120590

2)]Φ

= 0

(69)


119902 = minus

1

3 (120574 minus 1)

119901 =

3

2

1003816100381610038161003816120574 minus 1

1003816100381610038161003816

(70)


119911 =

2radic119887120574

3

1003816100381610038161003816120574 minus 1

1003816100381610038161003816

119890


A = 119911

minus(1198763(120574minus1))Φ(119911)

(71)



A120574= 119888

120574(

2radic119887120574

3

1003816100381610038161003816120574 minus 1

1003816100381610038161003816

119890


minus1198763(120574minus1)

sdot 119885119894](

2radic119887120574

3

1003816100381610038161003816120574 minus 1

1003816100381610038161003816

119890


(72)

with

] = plusmnradic1

9 (120574 minus 1)

2(4120590

2minus 119876

2) (73)



int

infin

minusinfin

sech(120587120578

2

) 119869119894120578(119911) 119889120578 = 2 sin (119911) (74)


Ψ (Ω 120573plusmn 120601 120590) = Φ (120601 120590)


3

1003816100381610038161003816120574 minus 1

1003816100381610038161003816

119890


sdot [1205781119890

1198941198862120573++ 120578

2119890

minus1198941198862120573+]

sdot [1198881119890

1198941198863120573minus+ 119888

2119890

minus1198941198863120573minus]

(75)



119889

2A1

119889Ω2minus 119876

119889A1

119889Ω

+ 120590

2

1A1= 0 120590

2

1= 119887

1+ 120590

2

(76)

whose solution is

A1= 119860

1119890

((119876+radic1198762+41205902

1)2)Ω

+ 1198602119890

((119876minusradic1198762+41205902

1)2)Ω

(77)


Ψ (Ω 120573plusmn 120601 120590) = Φ (120601 120590)

sdot [1198601119890

((119876+radic1198762+41205902

1)2)Ω

+ 1198602119890

((119876minusradic1198762+41205902

1)2)Ω

]

sdot [1205781119890

1198941198862120573++ 120578

2119890

minus1198941198862120573+] [119888

1119890

1198941198863120573minus+ 119888

2119890

minus1198941198863120573minus]

(78)


A = 119890

(1198762)Ω119885] (4radic

Λ

3

119890

3Ω) ] = plusmn

1

6

radic1198762+ 4120590

2

1 (79)


13


119889

2Aminus1

119889Ω2minus 119876

119889Aminus1

119889Ω

+ [119889minus1119890

6Ωminus 120590

2]A

minus1= 0

119889minus1= 48120583

minus1+ 48Λ

(80)

with the solution

Aminus1= (

radic119889minus1

3

119890

3Ω)

1198766

119885] (radic119889

minus1

3

119890

3Ω)

] = plusmn1

6

radic1198762+ 4120590

2

(81)




119889

2A0

119889Ω2+ (48Λ119890

6Ω+ 119887

0119890

3Ω+ 120590

2)A

0= 0 119887

0= 48120583

0(82)

with the solution

A0= 119890

minus3Ω2[119863

1119872


(

8119894119890

3ΩradicΛ

radic3

)


(

8119894119890

3ΩradicΛ

radic3

)]

(83)

where119872119896119901

And119882119896119901


tegration constants

6 Final Remarks







plusmn 120601 120590)Ψ =

119867119892(Ω 120573

plusmn)Ψ +

119867119898(120601 120590)Ψ = 0 where 119867

119892y 119867



plusmn) In order




Appendix



119866119872119873

= 119890

Φ2119866

119864

119872119873 (A1)


119878 =

1

21205812

10

int119889

10119883radicminus119866(119890

Φ2R + 4119866

119864119872119873nabla119872Φnabla

119873Φ)

+ int119889


119864119890

5Φ2Lmatt

(A2)


R = 119890

minusΦ2(R

119864minus

9

2

119866

119864119872119873nabla119872nabla119873Φ

minus

9

2

119866

119864119872119873nabla119872Φnabla

119873Φ)

(A3)


119878 =

1

21205812

10

int119889


119864(R

119864minus

9

2

119866

119864119872119873nabla119872nabla119873Φ

minus

1

2

119866

119864119872119873nabla119872Φnabla

119873Φ) + int119889


119864119890

5Φ2Lmatt

(A4)


det119866119872119873

=119866 = 119890

minus12120590 (A5)


R(10)

=R(4)

minus 42

120583]nabla120583120590nabla]120590 + 12

120583]nabla120583nabla]120590 (A6)

we obtain that

119878 =

1

21205812

10

int119889

4119909119889


minus6120590119890

minus2Φ[R

(4)

minus 42

120583]nabla120583120590nabla]120590 + 12

120583]nabla120583nabla]120590 + 4


(A7)


Φ = 120601 minus

1

2

ln (119881) (A8)


119878 =

1

21205812

10

int119889


minus2(120601+3120590)[R

(4)

minus 42

120583]nabla120583120590nabla]120590

+ 12

120583]nabla120583nabla]120590 + 4

120583]nabla120583120601nabla]120601]

(A9)





120583] = 119890

2Θ119892120583] (A10)


R(4)

= 119890

minus2Θ(R

(4)minus 6119892


120583]nabla120583Θnabla]Θ) (A11)


Θ = 120601 + 3120590 + ln(120581

2

10

1205812

4

) (A12)


119878 =

1

21205812

4

int119889



120583]nabla120583nabla]120590

minus 2119892



120583]nabla120583120601nabla]120590)

(A13)


int119889

10119883radicminus119866119890

minus2Φ120581

2

10119890

2Φ119879119872119873119866

119872119873

= int119889

4119909radicminus119892119890

2(Θ+120601)119879119872119873119866

119872119873

= int119889

4119909radicminus119892 (119890

2120601119879120583]119892

120583]+ 119890

2(Θ+120601)119879119898119899

119898119899)

(A14)


Competing Interests


Acknowledgments


References


























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






that

H119868vac=

1

8

119890

minus(1205731+1205732+1205733)[minusΠ

2

1minus Π

2

2minus Π

2

3+ 2Π

1Π2

+ 2Π1Π3+ 2Π

2Π3minus

16

11

Π

2

120601+

6

11

Π120601Π120590minus

1

33

Π

2

120590]

(24)



119875119894= 120597H120597119902

119894

become


119894= constant

Π120601= 0 997904rArr Π

120601= constant

Π120590= 0 997904rArr Π

120590= constant

120573

1015840

1=

1

4

119890

minus(1205731+1205732+1205733)[minusΠ

1+ Π

2+ Π

3]

120573

1015840

2=

1

4

119890

minus(1205731+1205732+1205733)[minusΠ

2+ Π

1+ Π

3]

120573

1015840

3=

1

4

119890

minus(1205731+1205732+1205733)[minusΠ

3+ Π

1+ Π

2]

120601

1015840= 120601

0119890

minus(1205731+1205732+1205733)

120590

1015840= 120590

0119890

minus(1205731+1205732+1205733)

(25)



1205731+ 120573

2+ 120573

3= ln [120598120591 + 119887

0] 120598 = 119887

1+ 119887

2+ 119887

3 (26)

with 119887119894= Π

119894and 119887



Ω = ln (120598120591 + 1198870)

13

120573+= ln (120598120591 + 119887

0)

(120598minus31198873)6120598

120573minus= ln (120598120591 + 119887

0)

(1198873+21198871minus120598)2radic3120598

(27)


120601 =

1206010

120598

ln (120598120591 + 1198870)

120590 =

1205900

120598

ln (120598120591 + 1198870)

(28)

where 1206010= Π

120601and 120590

0= Π




0


H119868=

1

8

119890

minus(1205731+1205732+1205733)[minusΠ

2

1minus Π

2

2minus Π

2

3+ 2Π

1Π2

+ 2Π1Π3+ 2Π

2Π3minus

16

11

Π

2

120601+

6

11

Π120601Π120590minus

1

33

Π

2

120590

minus 16Λ119890

2(1205731+1205732+1205733)]

(29)


Π

1015840

1= Π

1015840

2= Π

1015840

3= 4Λ119890

1205731+1205732+1205733

120573

1015840

1=

1

4

119890

minus(1205731+1205732+1205733)[minusΠ

1+ Π

2+ Π

3]

120573

1015840

2=

1

4

119890

minus(1205731+1205732+1205733)[minusΠ

2+ Π

1+ Π

3]

120573

1015840

3=

1

4

119890

minus(1205731+1205732+1205733)[minusΠ

3+ Π

1+ Π

2]

120601

1015840= 120601

0119890

minus(1205731+1205732+1205733)

120590

1015840= 120590

0119890

minus(1205731+1205732+1205733)

(30)




Π1= Π

2+ 120575

2= Π

3+ 120575

3 (31)

with 1205752and 120575



by

1

Λ

Π

10158402

1minus 3Π

2

1+ ]Π

1+ 120575

1= 0 (32)


] = 2 (1205752+ 120575

3)

1205751(1199020) = (120575

2minus 120575

3)

2

minus 1199020

(33)


1199020=

16

11

Π

2

120601minus

6

11

Π120601Π120590+

1

33

Π

2

120590 (34)


Π1=

]6

+

1

6



1205731+ 120573

2+ 120573

3= ln(1

8

radic

]2 + 121205751

3Λ

sinh (radic3Λ120591)) (36)



120601 =

81206012

radic]2 + 121205751

ln(tanh(radic3Λ

2

120591))

120590 =

81205902

radic]2 + 121205751

ln(tanh(radic3Λ

2

120591))

(37)



Ω =

1

3

ln(18

radic

]2 + 121205751

3Λ


120573+=

2

3

(1205752minus 2120575

3)

radic]2 + 121205751

ln tanh(radic3Λ

2

120591)

120573minus= minus

2

radic3

1205752

radic]2 + 121205751

ln tanh(radic3Λ

2

120591)

(38)


120588119886=

Λ

3

(1205752minus 2120575

3)

2

+ 120575

2

2

]2 + 12 ((1205752minus 120575

3)

2

minus 1199020)

coth2 (radic3Λ

2

120591) (39)









1and 120590

1 respectively

Π

1015840

1= Π

1015840

2= Π

1015840

3

= 4Λ119890

1205731+1205732+1205733+ 16120587119866 (1 minus 120574) 120588

120574119890

minus120574(1205731+1205732+1205733)

120573

1015840

1=

1

4

119890

minus(1205731+1205732+1205733)[minusΠ

1+ Π

2+ Π

3]

120573

1015840

2=

1

4

119890

minus(1205731+1205732+1205733)[minusΠ

2+ Π

1+ Π

3]

120573

1015840

3=

1

4

119890

minus(1205731+1205732+1205733)[minusΠ

3+ Π

1+ Π

2]

120601

1015840= 120601

2119890

minus(1205731+1205732+1205733)

120590

1015840= 120590

2119890

minus(1205731+1205732+1205733)

(40)


1 120573

2 and 120573

3


Π1= Π

2+ 119896

1= Π

3+ 119896

2 (41)


120601 = 1206012int 119890

minus(1205731+1205732+1205733)119889120591

120590 = 1205902int 119890

minus(1205731+1205732+1205733)119889120591

(42)



120582 =

4Λ 120574 = 1

4Λ + 32120587119866120588minus1 120574 = minus1

(43)

for whichΠ10158401= 120582119890



4

120582

Π

10158402

1minus 3Π

2

1+ ]Π

1+ 120575

1= 0 (44)


Π1=

]6

+

1

6

radic]2 + 121205751cosh(

radic3120582

2

120591) (45)


1205731+ 120573

2+ 120573

3= ln(1

4

radic

1205753+ 12120575

1

3120582

sinh(radic3120582

2

120591)) (46)


120601 =

81206012

radic]2 + 121205751


4

120591)

120590 =

81205902

radic]2 + 121205751


4

120591)

(47)



Ω =

1

3

ln(14

radic

120575

2+ 12120575

1

3120582

sinh(radic3120582

2

120591))

120573+=

2

3

(1205752minus 2120575

3)

radic]2 + 121205751


2

120591)

+

1

6

ln(14

radic

]2 + 121205751

3120582

)

120573minus= minus

2

radic3

1205752

radic]2 + 121205751


2

120591)

minus

1

2radic3

ln(14

radic

]2 + 121205751

3120582

)

(48)


08)119905 and 120590 = (120590

08)119905



Π

1015840

1= Π

1015840

2= Π

1015840

3= 4Λ119890

1205731+1205732+1205733+ 120572

0 120572

0= 16120587119866120588

0

Π

1015840

120601= 0 Π

120601= 120601

1= 119888119905119890

(49)


1

Λ

Π

10158402

1minus 3Π

2

1+ 120583Π

1+ 120585

1= 0 (50)

where

1205851= 120575

1minus

120572

2

0

Λ

(51)


Π1=

]6

+

1

6



1205731+ 120573

2+ 120573

3= ln(1

8

radic

]2 + 121205851

3Λ

sinh (radic3Λ120591)) (53)


120601 =

81206012

radic]2 + 121205851

ln tanh(radic3Λ

2

120591)

120590 =

81205902

radic]2 + 121205851

ln tanh(radic3Λ

2

120591)

(54)


Ω =

1

3

ln(18

radic

]2 + 121205851

3Λ


120573+=

2

3

1205752minus 2120575

3

radic]2 + 121205851

ln tanh(radic3Λ

2

120591)

120573minus= minus

2

radic3

1205752

radic]2 + 121205851

ln tanh(radic3Λ

2

120591)

(55)




0 In this




5 Quantum Scheme



+ 120573

minus)




minus3ΩΠ

2

Ωas

minus119890

minus(3minus119876)Ω120597Ω119890

minus119876Ω120597Ω= minus119890

minus3Ω120597

2

Ω+ 119876119890

minus3Ω120597Ω (56)


◻Ψ + 119876

120597Ψ

120597Ω

minus

48

11

120597

2Ψ

1205971206012minus

1

11

120597

2Ψ

1205971205902+

18

11

120597

2Ψ

120597120601120597120590

minus [119887120574119890

3(1minus120574)Ω+ 48Λ119890

6Ω]Ψ = 0

(57)



+ 120573



Ψ = Φ (120601 120590) 120595 (Ω 120573plusmn) (58)


◻120595 + 119876

120597120595

120597Ω

minus [119887120574119890

3(1minus120574)Ω+ 48Λ119890

6Ωminus

120583

2

5

]120595 = 0 (59)

48

11

120597

2Φ

1205971206012+

1

11

120597

2Φ

1205971205902minus

18

11

120597

2Φ

120597120601120597120590

+ 120583

2Φ = 0 (60)


Φ(120601 120590) = 1198621sin (119862

3120601 + 119862

4120590 + 119862

5)

+ 1198622cos (119862

3120601 + 119862

4120590 + 119862

5)

(61)

where 119862119894

5



plusmn 120601 120590)Ψ =

119867119892(Ω 120573

plusmn)Ψ +

119867119898(120601 120590)Ψ = 0 where

119867119892y 119867



plusmn) = A(Ω)B

1(120573

+)B

2(120573



119889

2A

119889Ω2minus 119876

119889A

119889Ω

+ [119887120574119890

minus3(120574minus1)Ω+ 48Λ119890

6Ω+ 120590

2]A = 0

119889

2B1

1198891205732

+

+ 119886

2

2B

1= 0 997904rArr

B1= 120578

1119890

1198941198862120573++ 120578

2119890

minus1198941198862120573+

119889

2B2

1198891205732

minus

+ 119886

2

3B

2= 0 997904rArr

B2= 119888

1119890

1198941198863120573minus+ 119888

2119890

minus1198941198863120573minus

(62)

where 1205902 = 11988621+ 119886

2

2+ 119886

2

3and 119888

119894and 120578




119889

2A

119889Ω2minus 119876

119889A

119889Ω

+ [119887120574119890

minus3(1minus120574)Ω+ 120590

2]A = 0 (63)


119911 =

radic119887120574

119901

119890


(64)


119889A

119889Ω

=

119889A

119889119911

119889119911

119889Ω

= minus

3

2

(120574 minus 1) 119911

119889A

119889119911

119889

2A

119889Ω2=

9

4

(120574 minus 1)

2

119911

2 1198892A

1198891199112+

9

4

(120574 minus 1)

2

119911

119889A

119889119911

(65)


9

4

(120574 minus 1)

2

119911

2 1198892A

1198891199112+

9

4

(120574 minus 1)

2

119911

119889A

119889119911

minus

3

2

119876 (120574 minus 1) 119911

119889A

119889119911

+ [119901

2119911

2+ 120590

2] = 0

(66)


A = 119911

119902119876Φ (119911) (67)


9

4

(120574 minus 1)

2

119911

119902119876[119911

2 1198892Φ

1198891199112+ 119911(1 + 2119902119876

+

2

3

119876

120574 minus 1

)

119889Φ

119889119911

+ (

4119901

2

9 (120574 minus 1)

2119911

2

+ 119876

2119902

2+

2

3

119902

120574 minus 1

+

4120590

2

9 (120574 minus 1)

2)Φ] = 0

(68)


119911

2 1198892Φ

1198891199112+ 119911

119889Φ

119889119911

+ [119911

2minus

1

9 (120574 minus 1)

2(119876

2minus 4120590

2)]Φ

= 0

(69)


119902 = minus

1

3 (120574 minus 1)

119901 =

3

2

1003816100381610038161003816120574 minus 1

1003816100381610038161003816

(70)


119911 =

2radic119887120574

3

1003816100381610038161003816120574 minus 1

1003816100381610038161003816

119890


A = 119911

minus(1198763(120574minus1))Φ(119911)

(71)



A120574= 119888

120574(

2radic119887120574

3

1003816100381610038161003816120574 minus 1

1003816100381610038161003816

119890


minus1198763(120574minus1)

sdot 119885119894](

2radic119887120574

3

1003816100381610038161003816120574 minus 1

1003816100381610038161003816

119890


(72)

with

] = plusmnradic1

9 (120574 minus 1)

2(4120590

2minus 119876

2) (73)



int

infin

minusinfin

sech(120587120578

2

) 119869119894120578(119911) 119889120578 = 2 sin (119911) (74)


Ψ (Ω 120573plusmn 120601 120590) = Φ (120601 120590)


3

1003816100381610038161003816120574 minus 1

1003816100381610038161003816

119890


sdot [1205781119890

1198941198862120573++ 120578

2119890

minus1198941198862120573+]

sdot [1198881119890

1198941198863120573minus+ 119888

2119890

minus1198941198863120573minus]

(75)



119889

2A1

119889Ω2minus 119876

119889A1

119889Ω

+ 120590

2

1A1= 0 120590

2

1= 119887

1+ 120590

2

(76)

whose solution is

A1= 119860

1119890

((119876+radic1198762+41205902

1)2)Ω

+ 1198602119890

((119876minusradic1198762+41205902

1)2)Ω

(77)


Ψ (Ω 120573plusmn 120601 120590) = Φ (120601 120590)

sdot [1198601119890

((119876+radic1198762+41205902

1)2)Ω

+ 1198602119890

((119876minusradic1198762+41205902

1)2)Ω

]

sdot [1205781119890

1198941198862120573++ 120578

2119890

minus1198941198862120573+] [119888

1119890

1198941198863120573minus+ 119888

2119890

minus1198941198863120573minus]

(78)


A = 119890

(1198762)Ω119885] (4radic

Λ

3

119890

3Ω) ] = plusmn

1

6

radic1198762+ 4120590

2

1 (79)


13


119889

2Aminus1

119889Ω2minus 119876

119889Aminus1

119889Ω

+ [119889minus1119890

6Ωminus 120590

2]A

minus1= 0

119889minus1= 48120583

minus1+ 48Λ

(80)

with the solution

Aminus1= (

radic119889minus1

3

119890

3Ω)

1198766

119885] (radic119889

minus1

3

119890

3Ω)

] = plusmn1

6

radic1198762+ 4120590

2

(81)




119889

2A0

119889Ω2+ (48Λ119890

6Ω+ 119887

0119890

3Ω+ 120590

2)A

0= 0 119887

0= 48120583

0(82)

with the solution

A0= 119890

minus3Ω2[119863

1119872


(

8119894119890

3ΩradicΛ

radic3

)


(

8119894119890

3ΩradicΛ

radic3

)]

(83)

where119872119896119901

And119882119896119901


tegration constants

6 Final Remarks







plusmn 120601 120590)Ψ =

119867119892(Ω 120573

plusmn)Ψ +

119867119898(120601 120590)Ψ = 0 where 119867

119892y 119867



plusmn) In order




Appendix



119866119872119873

= 119890

Φ2119866

119864

119872119873 (A1)


119878 =

1

21205812

10

int119889

10119883radicminus119866(119890

Φ2R + 4119866

119864119872119873nabla119872Φnabla

119873Φ)

+ int119889


119864119890

5Φ2Lmatt

(A2)


R = 119890

minusΦ2(R

119864minus

9

2

119866

119864119872119873nabla119872nabla119873Φ

minus

9

2

119866

119864119872119873nabla119872Φnabla

119873Φ)

(A3)


119878 =

1

21205812

10

int119889


119864(R

119864minus

9

2

119866

119864119872119873nabla119872nabla119873Φ

minus

1

2

119866

119864119872119873nabla119872Φnabla

119873Φ) + int119889


119864119890

5Φ2Lmatt

(A4)


det119866119872119873

=119866 = 119890

minus12120590 (A5)


R(10)

=R(4)

minus 42

120583]nabla120583120590nabla]120590 + 12

120583]nabla120583nabla]120590 (A6)

we obtain that

119878 =

1

21205812

10

int119889

4119909119889


minus6120590119890

minus2Φ[R

(4)

minus 42

120583]nabla120583120590nabla]120590 + 12

120583]nabla120583nabla]120590 + 4


(A7)


Φ = 120601 minus

1

2

ln (119881) (A8)


119878 =

1

21205812

10

int119889


minus2(120601+3120590)[R

(4)

minus 42

120583]nabla120583120590nabla]120590

+ 12

120583]nabla120583nabla]120590 + 4

120583]nabla120583120601nabla]120601]

(A9)





120583] = 119890

2Θ119892120583] (A10)


R(4)

= 119890

minus2Θ(R

(4)minus 6119892


120583]nabla120583Θnabla]Θ) (A11)


Θ = 120601 + 3120590 + ln(120581

2

10

1205812

4

) (A12)


119878 =

1

21205812

4

int119889



120583]nabla120583nabla]120590

minus 2119892



120583]nabla120583120601nabla]120590)

(A13)


int119889

10119883radicminus119866119890

minus2Φ120581

2

10119890

2Φ119879119872119873119866

119872119873

= int119889

4119909radicminus119892119890

2(Θ+120601)119879119872119873119866

119872119873

= int119889

4119909radicminus119892 (119890

2120601119879120583]119892

120583]+ 119890

2(Θ+120601)119879119898119899

119898119899)

(A14)


Competing Interests


Acknowledgments


References


























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





120601 =

81206012

radic]2 + 121205751

ln(tanh(radic3Λ

2

120591))

120590 =

81205902

radic]2 + 121205751

ln(tanh(radic3Λ

2

120591))

(37)



Ω =

1

3

ln(18

radic

]2 + 121205751

3Λ


120573+=

2

3

(1205752minus 2120575

3)

radic]2 + 121205751

ln tanh(radic3Λ

2

120591)

120573minus= minus

2

radic3

1205752

radic]2 + 121205751

ln tanh(radic3Λ

2

120591)

(38)


120588119886=

Λ

3

(1205752minus 2120575

3)

2

+ 120575

2

2

]2 + 12 ((1205752minus 120575

3)

2

minus 1199020)

coth2 (radic3Λ

2

120591) (39)









1and 120590

1 respectively

Π

1015840

1= Π

1015840

2= Π

1015840

3

= 4Λ119890

1205731+1205732+1205733+ 16120587119866 (1 minus 120574) 120588

120574119890

minus120574(1205731+1205732+1205733)

120573

1015840

1=

1

4

119890

minus(1205731+1205732+1205733)[minusΠ

1+ Π

2+ Π

3]

120573

1015840

2=

1

4

119890

minus(1205731+1205732+1205733)[minusΠ

2+ Π

1+ Π

3]

120573

1015840

3=

1

4

119890

minus(1205731+1205732+1205733)[minusΠ

3+ Π

1+ Π

2]

120601

1015840= 120601

2119890

minus(1205731+1205732+1205733)

120590

1015840= 120590

2119890

minus(1205731+1205732+1205733)

(40)


1 120573

2 and 120573

3


Π1= Π

2+ 119896

1= Π

3+ 119896

2 (41)


120601 = 1206012int 119890

minus(1205731+1205732+1205733)119889120591

120590 = 1205902int 119890

minus(1205731+1205732+1205733)119889120591

(42)



120582 =

4Λ 120574 = 1

4Λ + 32120587119866120588minus1 120574 = minus1

(43)

for whichΠ10158401= 120582119890



4

120582

Π

10158402

1minus 3Π

2

1+ ]Π

1+ 120575

1= 0 (44)


Π1=

]6

+

1

6

radic]2 + 121205751cosh(

radic3120582

2

120591) (45)


1205731+ 120573

2+ 120573

3= ln(1

4

radic

1205753+ 12120575

1

3120582

sinh(radic3120582

2

120591)) (46)


120601 =

81206012

radic]2 + 121205751


4

120591)

120590 =

81205902

radic]2 + 121205751


4

120591)

(47)



Ω =

1

3

ln(14

radic

120575

2+ 12120575

1

3120582

sinh(radic3120582

2

120591))

120573+=

2

3

(1205752minus 2120575

3)

radic]2 + 121205751


2

120591)

+

1

6

ln(14

radic

]2 + 121205751

3120582

)

120573minus= minus

2

radic3

1205752

radic]2 + 121205751


2

120591)

minus

1

2radic3

ln(14

radic

]2 + 121205751

3120582

)

(48)


08)119905 and 120590 = (120590

08)119905



Π

1015840

1= Π

1015840

2= Π

1015840

3= 4Λ119890

1205731+1205732+1205733+ 120572

0 120572

0= 16120587119866120588

0

Π

1015840

120601= 0 Π

120601= 120601

1= 119888119905119890

(49)


1

Λ

Π

10158402

1minus 3Π

2

1+ 120583Π

1+ 120585

1= 0 (50)

where

1205851= 120575

1minus

120572

2

0

Λ

(51)


Π1=

]6

+

1

6



1205731+ 120573

2+ 120573

3= ln(1

8

radic

]2 + 121205851

3Λ

sinh (radic3Λ120591)) (53)


120601 =

81206012

radic]2 + 121205851

ln tanh(radic3Λ

2

120591)

120590 =

81205902

radic]2 + 121205851

ln tanh(radic3Λ

2

120591)

(54)


Ω =

1

3

ln(18

radic

]2 + 121205851

3Λ


120573+=

2

3

1205752minus 2120575

3

radic]2 + 121205851

ln tanh(radic3Λ

2

120591)

120573minus= minus

2

radic3

1205752

radic]2 + 121205851

ln tanh(radic3Λ

2

120591)

(55)




0 In this




5 Quantum Scheme



+ 120573

minus)




minus3ΩΠ

2

Ωas

minus119890

minus(3minus119876)Ω120597Ω119890

minus119876Ω120597Ω= minus119890

minus3Ω120597

2

Ω+ 119876119890

minus3Ω120597Ω (56)


◻Ψ + 119876

120597Ψ

120597Ω

minus

48

11

120597

2Ψ

1205971206012minus

1

11

120597

2Ψ

1205971205902+

18

11

120597

2Ψ

120597120601120597120590

minus [119887120574119890

3(1minus120574)Ω+ 48Λ119890

6Ω]Ψ = 0

(57)



+ 120573



Ψ = Φ (120601 120590) 120595 (Ω 120573plusmn) (58)


◻120595 + 119876

120597120595

120597Ω

minus [119887120574119890

3(1minus120574)Ω+ 48Λ119890

6Ωminus

120583

2

5

]120595 = 0 (59)

48

11

120597

2Φ

1205971206012+

1

11

120597

2Φ

1205971205902minus

18

11

120597

2Φ

120597120601120597120590

+ 120583

2Φ = 0 (60)


Φ(120601 120590) = 1198621sin (119862

3120601 + 119862

4120590 + 119862

5)

+ 1198622cos (119862

3120601 + 119862

4120590 + 119862

5)

(61)

where 119862119894

5



plusmn 120601 120590)Ψ =

119867119892(Ω 120573

plusmn)Ψ +

119867119898(120601 120590)Ψ = 0 where

119867119892y 119867



plusmn) = A(Ω)B

1(120573

+)B

2(120573



119889

2A

119889Ω2minus 119876

119889A

119889Ω

+ [119887120574119890

minus3(120574minus1)Ω+ 48Λ119890

6Ω+ 120590

2]A = 0

119889

2B1

1198891205732

+

+ 119886

2

2B

1= 0 997904rArr

B1= 120578

1119890

1198941198862120573++ 120578

2119890

minus1198941198862120573+

119889

2B2

1198891205732

minus

+ 119886

2

3B

2= 0 997904rArr

B2= 119888

1119890

1198941198863120573minus+ 119888

2119890

minus1198941198863120573minus

(62)

where 1205902 = 11988621+ 119886

2

2+ 119886

2

3and 119888

119894and 120578




119889

2A

119889Ω2minus 119876

119889A

119889Ω

+ [119887120574119890

minus3(1minus120574)Ω+ 120590

2]A = 0 (63)


119911 =

radic119887120574

119901

119890


(64)


119889A

119889Ω

=

119889A

119889119911

119889119911

119889Ω

= minus

3

2

(120574 minus 1) 119911

119889A

119889119911

119889

2A

119889Ω2=

9

4

(120574 minus 1)

2

119911

2 1198892A

1198891199112+

9

4

(120574 minus 1)

2

119911

119889A

119889119911

(65)


9

4

(120574 minus 1)

2

119911

2 1198892A

1198891199112+

9

4

(120574 minus 1)

2

119911

119889A

119889119911

minus

3

2

119876 (120574 minus 1) 119911

119889A

119889119911

+ [119901

2119911

2+ 120590

2] = 0

(66)


A = 119911

119902119876Φ (119911) (67)


9

4

(120574 minus 1)

2

119911

119902119876[119911

2 1198892Φ

1198891199112+ 119911(1 + 2119902119876

+

2

3

119876

120574 minus 1

)

119889Φ

119889119911

+ (

4119901

2

9 (120574 minus 1)

2119911

2

+ 119876

2119902

2+

2

3

119902

120574 minus 1

+

4120590

2

9 (120574 minus 1)

2)Φ] = 0

(68)


119911

2 1198892Φ

1198891199112+ 119911

119889Φ

119889119911

+ [119911

2minus

1

9 (120574 minus 1)

2(119876

2minus 4120590

2)]Φ

= 0

(69)


119902 = minus

1

3 (120574 minus 1)

119901 =

3

2

1003816100381610038161003816120574 minus 1

1003816100381610038161003816

(70)


119911 =

2radic119887120574

3

1003816100381610038161003816120574 minus 1

1003816100381610038161003816

119890


A = 119911

minus(1198763(120574minus1))Φ(119911)

(71)



A120574= 119888

120574(

2radic119887120574

3

1003816100381610038161003816120574 minus 1

1003816100381610038161003816

119890


minus1198763(120574minus1)

sdot 119885119894](

2radic119887120574

3

1003816100381610038161003816120574 minus 1

1003816100381610038161003816

119890


(72)

with

] = plusmnradic1

9 (120574 minus 1)

2(4120590

2minus 119876

2) (73)



int

infin

minusinfin

sech(120587120578

2

) 119869119894120578(119911) 119889120578 = 2 sin (119911) (74)


Ψ (Ω 120573plusmn 120601 120590) = Φ (120601 120590)


3

1003816100381610038161003816120574 minus 1

1003816100381610038161003816

119890


sdot [1205781119890

1198941198862120573++ 120578

2119890

minus1198941198862120573+]

sdot [1198881119890

1198941198863120573minus+ 119888

2119890

minus1198941198863120573minus]

(75)



119889

2A1

119889Ω2minus 119876

119889A1

119889Ω

+ 120590

2

1A1= 0 120590

2

1= 119887

1+ 120590

2

(76)

whose solution is

A1= 119860

1119890

((119876+radic1198762+41205902

1)2)Ω

+ 1198602119890

((119876minusradic1198762+41205902

1)2)Ω

(77)


Ψ (Ω 120573plusmn 120601 120590) = Φ (120601 120590)

sdot [1198601119890

((119876+radic1198762+41205902

1)2)Ω

+ 1198602119890

((119876minusradic1198762+41205902

1)2)Ω

]

sdot [1205781119890

1198941198862120573++ 120578

2119890

minus1198941198862120573+] [119888

1119890

1198941198863120573minus+ 119888

2119890

minus1198941198863120573minus]

(78)


A = 119890

(1198762)Ω119885] (4radic

Λ

3

119890

3Ω) ] = plusmn

1

6

radic1198762+ 4120590

2

1 (79)


13


119889

2Aminus1

119889Ω2minus 119876

119889Aminus1

119889Ω

+ [119889minus1119890

6Ωminus 120590

2]A

minus1= 0

119889minus1= 48120583

minus1+ 48Λ

(80)

with the solution

Aminus1= (

radic119889minus1

3

119890

3Ω)

1198766

119885] (radic119889

minus1

3

119890

3Ω)

] = plusmn1

6

radic1198762+ 4120590

2

(81)




119889

2A0

119889Ω2+ (48Λ119890

6Ω+ 119887

0119890

3Ω+ 120590

2)A

0= 0 119887

0= 48120583

0(82)

with the solution

A0= 119890

minus3Ω2[119863

1119872


(

8119894119890

3ΩradicΛ

radic3

)


(

8119894119890

3ΩradicΛ

radic3

)]

(83)

where119872119896119901

And119882119896119901


tegration constants

6 Final Remarks







plusmn 120601 120590)Ψ =

119867119892(Ω 120573

plusmn)Ψ +

119867119898(120601 120590)Ψ = 0 where 119867

119892y 119867



plusmn) In order




Appendix



119866119872119873

= 119890

Φ2119866

119864

119872119873 (A1)


119878 =

1

21205812

10

int119889

10119883radicminus119866(119890

Φ2R + 4119866

119864119872119873nabla119872Φnabla

119873Φ)

+ int119889


119864119890

5Φ2Lmatt

(A2)


R = 119890

minusΦ2(R

119864minus

9

2

119866

119864119872119873nabla119872nabla119873Φ

minus

9

2

119866

119864119872119873nabla119872Φnabla

119873Φ)

(A3)


119878 =

1

21205812

10

int119889


119864(R

119864minus

9

2

119866

119864119872119873nabla119872nabla119873Φ

minus

1

2

119866

119864119872119873nabla119872Φnabla

119873Φ) + int119889


119864119890

5Φ2Lmatt

(A4)


det119866119872119873

=119866 = 119890

minus12120590 (A5)


R(10)

=R(4)

minus 42

120583]nabla120583120590nabla]120590 + 12

120583]nabla120583nabla]120590 (A6)

we obtain that

119878 =

1

21205812

10

int119889

4119909119889


minus6120590119890

minus2Φ[R

(4)

minus 42

120583]nabla120583120590nabla]120590 + 12

120583]nabla120583nabla]120590 + 4


(A7)


Φ = 120601 minus

1

2

ln (119881) (A8)


119878 =

1

21205812

10

int119889


minus2(120601+3120590)[R

(4)

minus 42

120583]nabla120583120590nabla]120590

+ 12

120583]nabla120583nabla]120590 + 4

120583]nabla120583120601nabla]120601]

(A9)





120583] = 119890

2Θ119892120583] (A10)


R(4)

= 119890

minus2Θ(R

(4)minus 6119892


120583]nabla120583Θnabla]Θ) (A11)


Θ = 120601 + 3120590 + ln(120581

2

10

1205812

4

) (A12)


119878 =

1

21205812

4

int119889



120583]nabla120583nabla]120590

minus 2119892



120583]nabla120583120601nabla]120590)

(A13)


int119889

10119883radicminus119866119890

minus2Φ120581

2

10119890

2Φ119879119872119873119866

119872119873

= int119889

4119909radicminus119892119890

2(Θ+120601)119879119872119873119866

119872119873

= int119889

4119909radicminus119892 (119890

2120601119879120583]119892

120583]+ 119890

2(Θ+120601)119879119898119899

119898119899)

(A14)


Competing Interests


Acknowledgments


References


























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





Ω =

1

3

ln(14

radic

120575

2+ 12120575

1

3120582

sinh(radic3120582

2

120591))

120573+=

2

3

(1205752minus 2120575

3)

radic]2 + 121205751


2

120591)

+

1

6

ln(14

radic

]2 + 121205751

3120582

)

120573minus= minus

2

radic3

1205752

radic]2 + 121205751


2

120591)

minus

1

2radic3

ln(14

radic

]2 + 121205751

3120582

)

(48)


08)119905 and 120590 = (120590

08)119905



Π

1015840

1= Π

1015840

2= Π

1015840

3= 4Λ119890

1205731+1205732+1205733+ 120572

0 120572

0= 16120587119866120588

0

Π

1015840

120601= 0 Π

120601= 120601

1= 119888119905119890

(49)


1

Λ

Π

10158402

1minus 3Π

2

1+ 120583Π

1+ 120585

1= 0 (50)

where

1205851= 120575

1minus

120572

2

0

Λ

(51)


Π1=

]6

+

1

6



1205731+ 120573

2+ 120573

3= ln(1

8

radic

]2 + 121205851

3Λ

sinh (radic3Λ120591)) (53)


120601 =

81206012

radic]2 + 121205851

ln tanh(radic3Λ

2

120591)

120590 =

81205902

radic]2 + 121205851

ln tanh(radic3Λ

2

120591)

(54)


Ω =

1

3

ln(18

radic

]2 + 121205851

3Λ


120573+=

2

3

1205752minus 2120575

3

radic]2 + 121205851

ln tanh(radic3Λ

2

120591)

120573minus= minus

2

radic3

1205752

radic]2 + 121205851

ln tanh(radic3Λ

2

120591)

(55)




0 In this




5 Quantum Scheme



+ 120573

minus)




minus3ΩΠ

2

Ωas

minus119890

minus(3minus119876)Ω120597Ω119890

minus119876Ω120597Ω= minus119890

minus3Ω120597

2

Ω+ 119876119890

minus3Ω120597Ω (56)


◻Ψ + 119876

120597Ψ

120597Ω

minus

48

11

120597

2Ψ

1205971206012minus

1

11

120597

2Ψ

1205971205902+

18

11

120597

2Ψ

120597120601120597120590

minus [119887120574119890

3(1minus120574)Ω+ 48Λ119890

6Ω]Ψ = 0

(57)



+ 120573



Ψ = Φ (120601 120590) 120595 (Ω 120573plusmn) (58)


◻120595 + 119876

120597120595

120597Ω

minus [119887120574119890

3(1minus120574)Ω+ 48Λ119890

6Ωminus

120583

2

5

]120595 = 0 (59)

48

11

120597

2Φ

1205971206012+

1

11

120597

2Φ

1205971205902minus

18

11

120597

2Φ

120597120601120597120590

+ 120583

2Φ = 0 (60)


Φ(120601 120590) = 1198621sin (119862

3120601 + 119862

4120590 + 119862

5)

+ 1198622cos (119862

3120601 + 119862

4120590 + 119862

5)

(61)

where 119862119894

5



plusmn 120601 120590)Ψ =

119867119892(Ω 120573

plusmn)Ψ +

119867119898(120601 120590)Ψ = 0 where

119867119892y 119867



plusmn) = A(Ω)B

1(120573

+)B

2(120573



119889

2A

119889Ω2minus 119876

119889A

119889Ω

+ [119887120574119890

minus3(120574minus1)Ω+ 48Λ119890

6Ω+ 120590

2]A = 0

119889

2B1

1198891205732

+

+ 119886

2

2B

1= 0 997904rArr

B1= 120578

1119890

1198941198862120573++ 120578

2119890

minus1198941198862120573+

119889

2B2

1198891205732

minus

+ 119886

2

3B

2= 0 997904rArr

B2= 119888

1119890

1198941198863120573minus+ 119888

2119890

minus1198941198863120573minus

(62)

where 1205902 = 11988621+ 119886

2

2+ 119886

2

3and 119888

119894and 120578




119889

2A

119889Ω2minus 119876

119889A

119889Ω

+ [119887120574119890

minus3(1minus120574)Ω+ 120590

2]A = 0 (63)


119911 =

radic119887120574

119901

119890


(64)


119889A

119889Ω

=

119889A

119889119911

119889119911

119889Ω

= minus

3

2

(120574 minus 1) 119911

119889A

119889119911

119889

2A

119889Ω2=

9

4

(120574 minus 1)

2

119911

2 1198892A

1198891199112+

9

4

(120574 minus 1)

2

119911

119889A

119889119911

(65)


9

4

(120574 minus 1)

2

119911

2 1198892A

1198891199112+

9

4

(120574 minus 1)

2

119911

119889A

119889119911

minus

3

2

119876 (120574 minus 1) 119911

119889A

119889119911

+ [119901

2119911

2+ 120590

2] = 0

(66)


A = 119911

119902119876Φ (119911) (67)


9

4

(120574 minus 1)

2

119911

119902119876[119911

2 1198892Φ

1198891199112+ 119911(1 + 2119902119876

+

2

3

119876

120574 minus 1

)

119889Φ

119889119911

+ (

4119901

2

9 (120574 minus 1)

2119911

2

+ 119876

2119902

2+

2

3

119902

120574 minus 1

+

4120590

2

9 (120574 minus 1)

2)Φ] = 0

(68)


119911

2 1198892Φ

1198891199112+ 119911

119889Φ

119889119911

+ [119911

2minus

1

9 (120574 minus 1)

2(119876

2minus 4120590

2)]Φ

= 0

(69)


119902 = minus

1

3 (120574 minus 1)

119901 =

3

2

1003816100381610038161003816120574 minus 1

1003816100381610038161003816

(70)


119911 =

2radic119887120574

3

1003816100381610038161003816120574 minus 1

1003816100381610038161003816

119890


A = 119911

minus(1198763(120574minus1))Φ(119911)

(71)



A120574= 119888

120574(

2radic119887120574

3

1003816100381610038161003816120574 minus 1

1003816100381610038161003816

119890


minus1198763(120574minus1)

sdot 119885119894](

2radic119887120574

3

1003816100381610038161003816120574 minus 1

1003816100381610038161003816

119890


(72)

with

] = plusmnradic1

9 (120574 minus 1)

2(4120590

2minus 119876

2) (73)



int

infin

minusinfin

sech(120587120578

2

) 119869119894120578(119911) 119889120578 = 2 sin (119911) (74)


Ψ (Ω 120573plusmn 120601 120590) = Φ (120601 120590)


3

1003816100381610038161003816120574 minus 1

1003816100381610038161003816

119890


sdot [1205781119890

1198941198862120573++ 120578

2119890

minus1198941198862120573+]

sdot [1198881119890

1198941198863120573minus+ 119888

2119890

minus1198941198863120573minus]

(75)



119889

2A1

119889Ω2minus 119876

119889A1

119889Ω

+ 120590

2

1A1= 0 120590

2

1= 119887

1+ 120590

2

(76)

whose solution is

A1= 119860

1119890

((119876+radic1198762+41205902

1)2)Ω

+ 1198602119890

((119876minusradic1198762+41205902

1)2)Ω

(77)


Ψ (Ω 120573plusmn 120601 120590) = Φ (120601 120590)

sdot [1198601119890

((119876+radic1198762+41205902

1)2)Ω

+ 1198602119890

((119876minusradic1198762+41205902

1)2)Ω

]

sdot [1205781119890

1198941198862120573++ 120578

2119890

minus1198941198862120573+] [119888

1119890

1198941198863120573minus+ 119888

2119890

minus1198941198863120573minus]

(78)


A = 119890

(1198762)Ω119885] (4radic

Λ

3

119890

3Ω) ] = plusmn

1

6

radic1198762+ 4120590

2

1 (79)


13


119889

2Aminus1

119889Ω2minus 119876

119889Aminus1

119889Ω

+ [119889minus1119890

6Ωminus 120590

2]A

minus1= 0

119889minus1= 48120583

minus1+ 48Λ

(80)

with the solution

Aminus1= (

radic119889minus1

3

119890

3Ω)

1198766

119885] (radic119889

minus1

3

119890

3Ω)

] = plusmn1

6

radic1198762+ 4120590

2

(81)




119889

2A0

119889Ω2+ (48Λ119890

6Ω+ 119887

0119890

3Ω+ 120590

2)A

0= 0 119887

0= 48120583

0(82)

with the solution

A0= 119890

minus3Ω2[119863

1119872


(

8119894119890

3ΩradicΛ

radic3

)


(

8119894119890

3ΩradicΛ

radic3

)]

(83)

where119872119896119901

And119882119896119901


tegration constants

6 Final Remarks







plusmn 120601 120590)Ψ =

119867119892(Ω 120573

plusmn)Ψ +

119867119898(120601 120590)Ψ = 0 where 119867

119892y 119867



plusmn) In order




Appendix



119866119872119873

= 119890

Φ2119866

119864

119872119873 (A1)


119878 =

1

21205812

10

int119889

10119883radicminus119866(119890

Φ2R + 4119866

119864119872119873nabla119872Φnabla

119873Φ)

+ int119889


119864119890

5Φ2Lmatt

(A2)


R = 119890

minusΦ2(R

119864minus

9

2

119866

119864119872119873nabla119872nabla119873Φ

minus

9

2

119866

119864119872119873nabla119872Φnabla

119873Φ)

(A3)


119878 =

1

21205812

10

int119889


119864(R

119864minus

9

2

119866

119864119872119873nabla119872nabla119873Φ

minus

1

2

119866

119864119872119873nabla119872Φnabla

119873Φ) + int119889


119864119890

5Φ2Lmatt

(A4)


det119866119872119873

=119866 = 119890

minus12120590 (A5)


R(10)

=R(4)

minus 42

120583]nabla120583120590nabla]120590 + 12

120583]nabla120583nabla]120590 (A6)

we obtain that

119878 =

1

21205812

10

int119889

4119909119889


minus6120590119890

minus2Φ[R

(4)

minus 42

120583]nabla120583120590nabla]120590 + 12

120583]nabla120583nabla]120590 + 4


(A7)


Φ = 120601 minus

1

2

ln (119881) (A8)


119878 =

1

21205812

10

int119889


minus2(120601+3120590)[R

(4)

minus 42

120583]nabla120583120590nabla]120590

+ 12

120583]nabla120583nabla]120590 + 4

120583]nabla120583120601nabla]120601]

(A9)





120583] = 119890

2Θ119892120583] (A10)


R(4)

= 119890

minus2Θ(R

(4)minus 6119892


120583]nabla120583Θnabla]Θ) (A11)


Θ = 120601 + 3120590 + ln(120581

2

10

1205812

4

) (A12)


119878 =

1

21205812

4

int119889



120583]nabla120583nabla]120590

minus 2119892



120583]nabla120583120601nabla]120590)

(A13)


int119889

10119883radicminus119866119890

minus2Φ120581

2

10119890

2Φ119879119872119873119866

119872119873

= int119889

4119909radicminus119892119890

2(Θ+120601)119879119872119873119866

119872119873

= int119889

4119909radicminus119892 (119890

2120601119879120583]119892

120583]+ 119890

2(Θ+120601)119879119898119899

119898119899)

(A14)


Competing Interests


Acknowledgments


References


























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





+ 120573



Ψ = Φ (120601 120590) 120595 (Ω 120573plusmn) (58)


◻120595 + 119876

120597120595

120597Ω

minus [119887120574119890

3(1minus120574)Ω+ 48Λ119890

6Ωminus

120583

2

5

]120595 = 0 (59)

48

11

120597

2Φ

1205971206012+

1

11

120597

2Φ

1205971205902minus

18

11

120597

2Φ

120597120601120597120590

+ 120583

2Φ = 0 (60)


Φ(120601 120590) = 1198621sin (119862

3120601 + 119862

4120590 + 119862

5)

+ 1198622cos (119862

3120601 + 119862

4120590 + 119862

5)

(61)

where 119862119894

5



plusmn 120601 120590)Ψ =

119867119892(Ω 120573

plusmn)Ψ +

119867119898(120601 120590)Ψ = 0 where

119867119892y 119867



plusmn) = A(Ω)B

1(120573

+)B

2(120573



119889

2A

119889Ω2minus 119876

119889A

119889Ω

+ [119887120574119890

minus3(120574minus1)Ω+ 48Λ119890

6Ω+ 120590

2]A = 0

119889

2B1

1198891205732

+

+ 119886

2

2B

1= 0 997904rArr

B1= 120578

1119890

1198941198862120573++ 120578

2119890

minus1198941198862120573+

119889

2B2

1198891205732

minus

+ 119886

2

3B

2= 0 997904rArr

B2= 119888

1119890

1198941198863120573minus+ 119888

2119890

minus1198941198863120573minus

(62)

where 1205902 = 11988621+ 119886

2

2+ 119886

2

3and 119888

119894and 120578




119889

2A

119889Ω2minus 119876

119889A

119889Ω

+ [119887120574119890

minus3(1minus120574)Ω+ 120590

2]A = 0 (63)


119911 =

radic119887120574

119901

119890


(64)


119889A

119889Ω

=

119889A

119889119911

119889119911

119889Ω

= minus

3

2

(120574 minus 1) 119911

119889A

119889119911

119889

2A

119889Ω2=

9

4

(120574 minus 1)

2

119911

2 1198892A

1198891199112+

9

4

(120574 minus 1)

2

119911

119889A

119889119911

(65)


9

4

(120574 minus 1)

2

119911

2 1198892A

1198891199112+

9

4

(120574 minus 1)

2

119911

119889A

119889119911

minus

3

2

119876 (120574 minus 1) 119911

119889A

119889119911

+ [119901

2119911

2+ 120590

2] = 0

(66)


A = 119911

119902119876Φ (119911) (67)


9

4

(120574 minus 1)

2

119911

119902119876[119911

2 1198892Φ

1198891199112+ 119911(1 + 2119902119876

+

2

3

119876

120574 minus 1

)

119889Φ

119889119911

+ (

4119901

2

9 (120574 minus 1)

2119911

2

+ 119876

2119902

2+

2

3

119902

120574 minus 1

+

4120590

2

9 (120574 minus 1)

2)Φ] = 0

(68)


119911

2 1198892Φ

1198891199112+ 119911

119889Φ

119889119911

+ [119911

2minus

1

9 (120574 minus 1)

2(119876

2minus 4120590

2)]Φ

= 0

(69)


119902 = minus

1

3 (120574 minus 1)

119901 =

3

2

1003816100381610038161003816120574 minus 1

1003816100381610038161003816

(70)


119911 =

2radic119887120574

3

1003816100381610038161003816120574 minus 1

1003816100381610038161003816

119890


A = 119911

minus(1198763(120574minus1))Φ(119911)

(71)



A120574= 119888

120574(

2radic119887120574

3

1003816100381610038161003816120574 minus 1

1003816100381610038161003816

119890


minus1198763(120574minus1)

sdot 119885119894](

2radic119887120574

3

1003816100381610038161003816120574 minus 1

1003816100381610038161003816

119890


(72)

with

] = plusmnradic1

9 (120574 minus 1)

2(4120590

2minus 119876

2) (73)



int

infin

minusinfin

sech(120587120578

2

) 119869119894120578(119911) 119889120578 = 2 sin (119911) (74)


Ψ (Ω 120573plusmn 120601 120590) = Φ (120601 120590)


3

1003816100381610038161003816120574 minus 1

1003816100381610038161003816

119890


sdot [1205781119890

1198941198862120573++ 120578

2119890

minus1198941198862120573+]

sdot [1198881119890

1198941198863120573minus+ 119888

2119890

minus1198941198863120573minus]

(75)



119889

2A1

119889Ω2minus 119876

119889A1

119889Ω

+ 120590

2

1A1= 0 120590

2

1= 119887

1+ 120590

2

(76)

whose solution is

A1= 119860

1119890

((119876+radic1198762+41205902

1)2)Ω

+ 1198602119890

((119876minusradic1198762+41205902

1)2)Ω

(77)


Ψ (Ω 120573plusmn 120601 120590) = Φ (120601 120590)

sdot [1198601119890

((119876+radic1198762+41205902

1)2)Ω

+ 1198602119890

((119876minusradic1198762+41205902

1)2)Ω

]

sdot [1205781119890

1198941198862120573++ 120578

2119890

minus1198941198862120573+] [119888

1119890

1198941198863120573minus+ 119888

2119890

minus1198941198863120573minus]

(78)


A = 119890

(1198762)Ω119885] (4radic

Λ

3

119890

3Ω) ] = plusmn

1

6

radic1198762+ 4120590

2

1 (79)


13


119889

2Aminus1

119889Ω2minus 119876

119889Aminus1

119889Ω

+ [119889minus1119890

6Ωminus 120590

2]A

minus1= 0

119889minus1= 48120583

minus1+ 48Λ

(80)

with the solution

Aminus1= (

radic119889minus1

3

119890

3Ω)

1198766

119885] (radic119889

minus1

3

119890

3Ω)

] = plusmn1

6

radic1198762+ 4120590

2

(81)




119889

2A0

119889Ω2+ (48Λ119890

6Ω+ 119887

0119890

3Ω+ 120590

2)A

0= 0 119887

0= 48120583

0(82)

with the solution

A0= 119890

minus3Ω2[119863

1119872


(

8119894119890

3ΩradicΛ

radic3

)


(

8119894119890

3ΩradicΛ

radic3

)]

(83)

where119872119896119901

And119882119896119901


tegration constants

6 Final Remarks







plusmn 120601 120590)Ψ =

119867119892(Ω 120573

plusmn)Ψ +

119867119898(120601 120590)Ψ = 0 where 119867

119892y 119867



plusmn) In order




Appendix



119866119872119873

= 119890

Φ2119866

119864

119872119873 (A1)


119878 =

1

21205812

10

int119889

10119883radicminus119866(119890

Φ2R + 4119866

119864119872119873nabla119872Φnabla

119873Φ)

+ int119889


119864119890

5Φ2Lmatt

(A2)


R = 119890

minusΦ2(R

119864minus

9

2

119866

119864119872119873nabla119872nabla119873Φ

minus

9

2

119866

119864119872119873nabla119872Φnabla

119873Φ)

(A3)


119878 =

1

21205812

10

int119889


119864(R

119864minus

9

2

119866

119864119872119873nabla119872nabla119873Φ

minus

1

2

119866

119864119872119873nabla119872Φnabla

119873Φ) + int119889


119864119890

5Φ2Lmatt

(A4)


det119866119872119873

=119866 = 119890

minus12120590 (A5)


R(10)

=R(4)

minus 42

120583]nabla120583120590nabla]120590 + 12

120583]nabla120583nabla]120590 (A6)

we obtain that

119878 =

1

21205812

10

int119889

4119909119889


minus6120590119890

minus2Φ[R

(4)

minus 42

120583]nabla120583120590nabla]120590 + 12

120583]nabla120583nabla]120590 + 4


(A7)


Φ = 120601 minus

1

2

ln (119881) (A8)


119878 =

1

21205812

10

int119889


minus2(120601+3120590)[R

(4)

minus 42

120583]nabla120583120590nabla]120590

+ 12

120583]nabla120583nabla]120590 + 4

120583]nabla120583120601nabla]120601]

(A9)





120583] = 119890

2Θ119892120583] (A10)


R(4)

= 119890

minus2Θ(R

(4)minus 6119892


120583]nabla120583Θnabla]Θ) (A11)


Θ = 120601 + 3120590 + ln(120581

2

10

1205812

4

) (A12)


119878 =

1

21205812

4

int119889



120583]nabla120583nabla]120590

minus 2119892



120583]nabla120583120601nabla]120590)

(A13)


int119889

10119883radicminus119866119890

minus2Φ120581

2

10119890

2Φ119879119872119873119866

119872119873

= int119889

4119909radicminus119892119890

2(Θ+120601)119879119872119873119866

119872119873

= int119889

4119909radicminus119892 (119890

2120601119879120583]119892

120583]+ 119890

2(Θ+120601)119879119898119899

119898119899)

(A14)


Competing Interests


Acknowledgments


References


























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





A120574= 119888

120574(

2radic119887120574

3

1003816100381610038161003816120574 minus 1

1003816100381610038161003816

119890


minus1198763(120574minus1)

sdot 119885119894](

2radic119887120574

3

1003816100381610038161003816120574 minus 1

1003816100381610038161003816

119890


(72)

with

] = plusmnradic1

9 (120574 minus 1)

2(4120590

2minus 119876

2) (73)



int

infin

minusinfin

sech(120587120578

2

) 119869119894120578(119911) 119889120578 = 2 sin (119911) (74)


Ψ (Ω 120573plusmn 120601 120590) = Φ (120601 120590)


3

1003816100381610038161003816120574 minus 1

1003816100381610038161003816

119890


sdot [1205781119890

1198941198862120573++ 120578

2119890

minus1198941198862120573+]

sdot [1198881119890

1198941198863120573minus+ 119888

2119890

minus1198941198863120573minus]

(75)



119889

2A1

119889Ω2minus 119876

119889A1

119889Ω

+ 120590

2

1A1= 0 120590

2

1= 119887

1+ 120590

2

(76)

whose solution is

A1= 119860

1119890

((119876+radic1198762+41205902

1)2)Ω

+ 1198602119890

((119876minusradic1198762+41205902

1)2)Ω

(77)


Ψ (Ω 120573plusmn 120601 120590) = Φ (120601 120590)

sdot [1198601119890

((119876+radic1198762+41205902

1)2)Ω

+ 1198602119890

((119876minusradic1198762+41205902

1)2)Ω

]

sdot [1205781119890

1198941198862120573++ 120578

2119890

minus1198941198862120573+] [119888

1119890

1198941198863120573minus+ 119888

2119890

minus1198941198863120573minus]

(78)


A = 119890

(1198762)Ω119885] (4radic

Λ

3

119890

3Ω) ] = plusmn

1

6

radic1198762+ 4120590

2

1 (79)


13


119889

2Aminus1

119889Ω2minus 119876

119889Aminus1

119889Ω

+ [119889minus1119890

6Ωminus 120590

2]A

minus1= 0

119889minus1= 48120583

minus1+ 48Λ

(80)

with the solution

Aminus1= (

radic119889minus1

3

119890

3Ω)

1198766

119885] (radic119889

minus1

3

119890

3Ω)

] = plusmn1

6

radic1198762+ 4120590

2

(81)




119889

2A0

119889Ω2+ (48Λ119890

6Ω+ 119887

0119890

3Ω+ 120590

2)A

0= 0 119887

0= 48120583

0(82)

with the solution

A0= 119890

minus3Ω2[119863

1119872


(

8119894119890

3ΩradicΛ

radic3

)


(

8119894119890

3ΩradicΛ

radic3

)]

(83)

where119872119896119901

And119882119896119901


tegration constants

6 Final Remarks







plusmn 120601 120590)Ψ =

119867119892(Ω 120573

plusmn)Ψ +

119867119898(120601 120590)Ψ = 0 where 119867

119892y 119867



plusmn) In order




Appendix



119866119872119873

= 119890

Φ2119866

119864

119872119873 (A1)


119878 =

1

21205812

10

int119889

10119883radicminus119866(119890

Φ2R + 4119866

119864119872119873nabla119872Φnabla

119873Φ)

+ int119889


119864119890

5Φ2Lmatt

(A2)


R = 119890

minusΦ2(R

119864minus

9

2

119866

119864119872119873nabla119872nabla119873Φ

minus

9

2

119866

119864119872119873nabla119872Φnabla

119873Φ)

(A3)


119878 =

1

21205812

10

int119889


119864(R

119864minus

9

2

119866

119864119872119873nabla119872nabla119873Φ

minus

1

2

119866

119864119872119873nabla119872Φnabla

119873Φ) + int119889


119864119890

5Φ2Lmatt

(A4)


det119866119872119873

=119866 = 119890

minus12120590 (A5)


R(10)

=R(4)

minus 42

120583]nabla120583120590nabla]120590 + 12

120583]nabla120583nabla]120590 (A6)

we obtain that

119878 =

1

21205812

10

int119889

4119909119889


minus6120590119890

minus2Φ[R

(4)

minus 42

120583]nabla120583120590nabla]120590 + 12

120583]nabla120583nabla]120590 + 4


(A7)


Φ = 120601 minus

1

2

ln (119881) (A8)


119878 =

1

21205812

10

int119889


minus2(120601+3120590)[R

(4)

minus 42

120583]nabla120583120590nabla]120590

+ 12

120583]nabla120583nabla]120590 + 4

120583]nabla120583120601nabla]120601]

(A9)





120583] = 119890

2Θ119892120583] (A10)


R(4)

= 119890

minus2Θ(R

(4)minus 6119892


120583]nabla120583Θnabla]Θ) (A11)


Θ = 120601 + 3120590 + ln(120581

2

10

1205812

4

) (A12)


119878 =

1

21205812

4

int119889



120583]nabla120583nabla]120590

minus 2119892



120583]nabla120583120601nabla]120590)

(A13)


int119889

10119883radicminus119866119890

minus2Φ120581

2

10119890

2Φ119879119872119873119866

119872119873

= int119889

4119909radicminus119892119890

2(Θ+120601)119879119872119873119866

119872119873

= int119889

4119909radicminus119892 (119890

2120601119879120583]119892

120583]+ 119890

2(Θ+120601)119879119898119899

119898119899)

(A14)


Competing Interests


Acknowledgments


References


























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







plusmn 120601 120590)Ψ =

119867119892(Ω 120573

plusmn)Ψ +

119867119898(120601 120590)Ψ = 0 where 119867

119892y 119867



plusmn) In order




Appendix



119866119872119873

= 119890

Φ2119866

119864

119872119873 (A1)


119878 =

1

21205812

10

int119889

10119883radicminus119866(119890

Φ2R + 4119866

119864119872119873nabla119872Φnabla

119873Φ)

+ int119889


119864119890

5Φ2Lmatt

(A2)


R = 119890

minusΦ2(R

119864minus

9

2

119866

119864119872119873nabla119872nabla119873Φ

minus

9

2

119866

119864119872119873nabla119872Φnabla

119873Φ)

(A3)


119878 =

1

21205812

10

int119889


119864(R

119864minus

9

2

119866

119864119872119873nabla119872nabla119873Φ

minus

1

2

119866

119864119872119873nabla119872Φnabla

119873Φ) + int119889


119864119890

5Φ2Lmatt

(A4)


det119866119872119873

=119866 = 119890

minus12120590 (A5)


R(10)

=R(4)

minus 42

120583]nabla120583120590nabla]120590 + 12

120583]nabla120583nabla]120590 (A6)

we obtain that

119878 =

1

21205812

10

int119889

4119909119889


minus6120590119890

minus2Φ[R

(4)

minus 42

120583]nabla120583120590nabla]120590 + 12

120583]nabla120583nabla]120590 + 4


(A7)


Φ = 120601 minus

1

2

ln (119881) (A8)


119878 =

1

21205812

10

int119889


minus2(120601+3120590)[R

(4)

minus 42

120583]nabla120583120590nabla]120590

+ 12

120583]nabla120583nabla]120590 + 4

120583]nabla120583120601nabla]120601]

(A9)





120583] = 119890

2Θ119892120583] (A10)


R(4)

= 119890

minus2Θ(R

(4)minus 6119892


120583]nabla120583Θnabla]Θ) (A11)


Θ = 120601 + 3120590 + ln(120581

2

10

1205812

4

) (A12)


119878 =

1

21205812

4

int119889



120583]nabla120583nabla]120590

minus 2119892



120583]nabla120583120601nabla]120590)

(A13)


int119889

10119883radicminus119866119890

minus2Φ120581

2

10119890

2Φ119879119872119873119866

119872119873

= int119889

4119909radicminus119892119890

2(Θ+120601)119879119872119873119866

119872119873

= int119889

4119909radicminus119892 (119890

2120601119879120583]119892

120583]+ 119890

2(Θ+120601)119879119898119899

119898119899)

(A14)


Competing Interests


Acknowledgments


References


























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







120583] = 119890

2Θ119892120583] (A10)


R(4)

= 119890

minus2Θ(R

(4)minus 6119892


120583]nabla120583Θnabla]Θ) (A11)


Θ = 120601 + 3120590 + ln(120581

2

10

1205812

4

) (A12)


119878 =

1

21205812

4

int119889



120583]nabla120583nabla]120590

minus 2119892



120583]nabla120583120601nabla]120590)

(A13)


int119889

10119883radicminus119866119890

minus2Φ120581

2

10119890

2Φ119879119872119873119866

119872119873

= int119889

4119909radicminus119892119890

2(Θ+120601)119879119872119873119866

119872119873

= int119889

4119909radicminus119892 (119890

2120601119879120583]119892

120583]+ 119890

2(Θ+120601)119879119898119899

119898119899)

(A14)


Competing Interests


Acknowledgments


References


























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics









































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



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