Research Article Geometric Methods to Investigate ...downloads.hindawi.com/journals/ijmms/2013/504645.pdfGeometric approaches have been found useful in producing a great variety of

Hindawi Publishing CorporationInternational Journal of Mathematics and Mathematical SciencesVolume 2013 Article ID 504645 8 pageshttpdxdoiorg1011552013504645

Research ArticleGeometric Methods to Investigate Prolongation Structures forDifferential Systems with Applications to Integrable Systems

Paul Bracken

Department of Mathematics University of Texas Edinburg TX 78541-2999 USA

Correspondence should be addressed to Paul Bracken brackenutpaedu

Received 8 January 2013 Accepted 13 February 2013

Academic Editor Aloys Krieg

Copyright copy 2013 Paul BrackenThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A type of prolongation structure for several general systems is discussed They are based on a set of one forms for which theunderlying structure group of the integrability condition corresponds to the Lie algebra of SL(2 R) O(3) and SU(3) Each will beconsidered in turn and the latter two systems represent larger 3 times 3 cases This geometric approach is applied to all of the three ofthese systems to obtain prolongation structures explicitly In both 3 times 3 cases the prolongation structure is reduced to the situationof three smaller 2 times 2 problems

1 Introduction

Geometric approaches have been found useful in producinga great variety of results for nonlinear partial differentialequations [1] A specific geometric approach discussed herehas been found to produce a very elegant coherent andunified understanding of many ideas in nonlinear physicsby means of fundamental differential geometric conceptsIn fact relationships between a geometric interpretationof soliton equations prolongation structure Lax pairs andconservation laws can be clearly realized and made use ofThe interest in the approach its generality and the resultsit produces do not depend on a specific equation at theoutset The formalism in terms of differential forms [2]can encompass large classes of nonlinear partial differentialequation certainly the AKNS systems [3 4] and it allowsthe production of generic expressions for infinite numbers ofconservation laws Moreover it leads to the consequence thatmany seemingly different equations turn out to be related bya gauge transformation

Here the discussion begins by studying prolongationstructures for a 2 times 2 119878119871(2R) system discussed first by Sasaki[5 6] and Crampin [7] to present and illustrate the methodThis will also demonstrate the procedure and the kind ofprolongation results that emerge It also provides a basis fromwhich to work out larger systems since they can generally be

reduced to 2 times 2 problems Of greater complexity are a pairof 3 times 3 problems which will be considered next It is shownhow to construct an 119874(3) system based on three constituentone forms as well as an 119878119880(3) system composed of eightfundamental one formsThe former has not appearedThe 3times3 problem is less well known than the 2times2 problem howeversomework has appeared in [8 9] In the first of these the 3times3solitons degenerate to the AKNS solitons in three ways whilein the latter nondegenerate case they do not All of the resultsare presented explicitly that is the coefficients of all theforms and their higher exterior derivatives are calculated andgiven explicitly Maple is used to do this in the harder casesThe approach is quite unified and so once the formalism isestablished for the 119878119871(2R) system the overall procedure canbe carried over to the other Lie algebra cases as well

Of course the 3 times 3 problems will yield more types ofconservation laws The prolongation structures of the 3 times 3

problems will be observed to be reduceable to the same typeas the 2 times 2 system considered at the beginning Howeverthe Riccati representations become much more complicated[10]The generalization of this formalism to an 119899times119899 problemthen becomes straightforward after completing the study ofthese smaller cases [11]The prolongation structure of an 119899times119899problem is reduced to a collection of 119899 smaller (119899minus1)times(119899minus1)cases and finally at the end to a set of 2 times 2 problems such asthe 119878119871(2R) case at the beginning Finally to summarize at

2 International Journal of Mathematics and Mathematical Sciences

the end a collection of conservation laws are developed fromone of the results and some speculation as to how this typerelates to other procedures [12 13]

2 119878119871(2R) Prolongations

The procedure begins by associating a system of Pfaffianequations to the nonlinear system to be studied namely

120572119894= 0 120572

119894= 119889119910119894minus Ω119894119895119910119895 119894 119895 = 1 2 (1)

In (1) Ω is a traceless 2 times 2 matrix which consists of afamily of one forms 120596

119894 At the end it is desired to express

these one forms in terms of independent variables which arecalled 119909 and 119905 the dependent variables and their derivativesHowever no specific equation need be assumed at the startThe underlying structure group is 119878119871(2R) Explicitly thematrix of one forms is given by the 2 times 2matrix

Ω = (1205961

1205962

1205963minus1205961

) (2)

The integrability conditions are expressed as the vanishing ofa traceless 2 times 2matrix of two forms Θ given by

Θ = 119889Ω minus Ω and Ω (3)

In terms of components the left-hand side of (3) has the form

Θ = (Θ119894119895) = (

1205991

1205992

1205993minus1205991

) (4)

Substituting (2) into (3) a matrix which contains the integra-bility equations in terms of the basis one forms is obtainedThe three equations are

1205991= 1198891205961minus 1205962and 1205963 120599

2= 1198891205962minus 21205961and 1205962

1205993= 1198891205963+ 21205961and 1205963

(5)

Of use in the theorems which follow it is also useful to havethe 119889120596

119894in (5) expressed explicitly in terms of the 120599

119894as

1198891205961= 1205991+ 1205962and 1205963 119889120596

2= 1205992+ 21205961and 1205962

1198891205963= 1205993minus 21205961and 1205963

(6)

Therefore by selecting a particular choice for the set 120596119894 the

nonlinear equation of interest can be written simply as

Θ = 0 120599119894= 0 119894 = 1 2 3 (7)

It is easy to see that the system is closed Upon differentiatingΘ and substituting (3) we obtain

119889Θ = Ω and Θ minus Θ and Ω (8)

This implies that the exterior derivatives of the set of 120599119894 are

contained in the ring of two forms 120599119894

Now the differential ideal can be prolonged by includingthe forms 120572

1 1205722 given by (1)

Theorem 1 The differential one forms

1205721= 1198891199101minus 12059611199101minus 12059621199102 120572

2= 1198891199102minus 12059631199101+ 12059611199102 (9)

have the following exterior derivatives

1198891205721= 1205961and 1205721+ 1205962and 1205722minus 11991011205991minus 11991021205992

1198891205722= minus 120596

1and 1205722+ 1205963and 1205721+ 11991021205991minus 11991011205993

(10)

Hence the exterior derivatives of the 120572119894 are contained in the

ring of forms 120599119894 120572119895

Proof The forms (9) follow by substituting (2) into (1)Differentiating 120572

1in (9) it is found that

1198891205721= minus11991011198891205961+ 1205961and 1198891199101minus 11991021198891205962+ 1205962and 1198891199102 (11)

Obtaining 1198891199101and 119889119910

2from (9) and 119889120596

1 1198891205962from (6) 119889120572

1

becomes

1198891205721= minus11991011205991minus 11991011205962and 1205963+ 1205961and 1205721+ 11991021205961and 1205962minus 11991021205992

minus 211991021205961and 1205962+ 1205962and 1205722+ 11991011205962and 1205963minus 11991021205962and 1205961

= 1205961and 1205721+ 1205962and 1205722minus 11991011205991minus 11991021205992

(12)

Similarly beginning with 1205722

1198891205722= minus11991011198891205963+ 1205963and 1198891199101+ 11991021198891205961minus 1205961and 1198891199102

= minus11991011205993+ 211991011205961and 1205963+ 1205963and 1205721

+ 11991011205963and 1205961+ 11991021205963and 1205962

+ 11991021205991+ 11991021205962and 1205963minus 1205961and 1205722minus 11991011205961and 1205963

= minus1205961and 1205722+ 1205963and 1205721+ 11991021205991minus 11991011205993

(13)

as required

Corollary 2 The exterior derivatives of the 120572119894in (9) can be

expressed concisely in terms of the matrix elements ofΘ andΩas

119889120572119894= minusΘ119894119895119910119895+ Ω119894119895and 120572119895 119894 119895 = 1 2 (14)

The corollary is Theorem 1 after using the definitions ofthe matrices in (2) and (4)

The forms 1205721 1205722given in (9) lead to a natural Riccati

representation called 1205723 1205724for this differential system This

arises by taking the following linear combinations of 1205721and

1205722

1199102

11205723= 11991011205722minus 11991021205721

= 11991011198891199102minus 11991021198891199101minus 1199102

11205963+ 2119910111991021205961+ 1199102

21205962

1199102

21205724= 11991021205721minus 11991011205722

= 11991021198891199101minus 11991011198891199102minus 1199102

21205962minus 2119910111991021205961+ 1199102

11205963

(15)

International Journal of Mathematics and Mathematical Sciences 3

Two new functions or pseudopotentials 1199103and 119910

4can be

introduced which are defined by the transformation

1199103=1199102

1199101

1199104=1199101

1199102

(16)

Then from (15) the following forms which have a Riccatistructure result

1205723= 1198891199103minus 1205963+ 211991031205961+ 1199102

31205962

1205724= 1198891199104minus 1205962minus 211991041205961+ 1199102

41205963

(17)

These equations could also be thought of as Riccati equationsfor 1199103and 119910

4

Theorem3 (i)Theone forms12057231205724given by (17) have exterior

derivatives

1198891205723= 211991031205991+ 1199102

31205992minus 1205993+ 21205723and (1205961+ 11991031205962)

1198891205724= minus 2119910

41205991minus 1205992+ 1199102

41205993+ 21205724and (minus120596

1+ 11991041205963)

(18)

Consequently results (18) are contained in the ring spanned by120599119894 and 120572

119895

(ii) Define the forms 1205901= 1205961+ 11991031205962and 120590

2= minus120596

1+

11991041205963appearing in (18) The exterior derivatives of 120590

1and 120590

2

are found to be

1198891205901= 1205991+ 11991031205992+ 1205723and 1205962

1198891205902= minus1205991+ 11991031205992+ 1205724and 1205962

(19)

Therefore 1198891205901equiv 0 and 119889120590

2equiv 0 mod 120599

119894 120572119895

The proof of this Theorem proceeds along exactly thesame lines as Theorem 1 The 119878119871(2R) structure and theconnection interpretation can be based on the forms 120572

3 1205724

At this point the process can be continued Two morepseudopotentials 119910

5and 119910

6can be introduced and the

differential system can be extended by including two moreone forms 120572

5and 120572

6 The specific structure of these forms is

suggested byTheorem 3

Theorem 4 Define two one forms 1205725and 120572

6as

1205725= 1198891199105minus 1205961minus 11991031205962 120572

6= 1198891199106+ 1205961minus 11991041205963 (20)

The exterior derivatives of the one forms (20) are given by theexpressions

1198891205725= minus1205991minus 11991031205993minus 1205723and 1205962 119889120572

6= 1205991minus 11991041205993minus 1205724and 1205963

(21)

Results (21) specify the closure properties of the forms 1205725and

1205726 This theorem is proved along similar lines keeping in mind

that the results for 1198891199103and 119889119910

4are obtained from (17)

A final extension can be made by adding two additionalpseudopotentials119910

7and1199108 At each stage the ideal of forms is

being enlarged and is found to be closed over the underlyingsubideal which does not contain the two new forms

Theorem 5 Define the one forms 1205727and 120572

8by means of

1205727= 1198891199107minus 119890minus211991051205962 120572

8= 1198891199108minus 119890minus211991061205963 (22)

The exterior derivatives of the forms 1205727and 120572

8are given by

1198891205727= 2119890minus211991051205725and 1205962minus 119890minus211991051205992


(23)

and are closed over the prolonged ideal

3 119874(3) Prolongations

The first 3 times 3 problem we consider is formulated in terms ofa set of one forms 120596

119894for 119894 = 1 2 3 The following system of

Pfaffian equations is to be associated to a specific nonlinearequation

120572119894= 0 120572

119894= 119889119910119894minus Ω119894119895119910119895 119894 119895 = 1 2 3 (24)

In (24) Ω is a traceless 3 times 3 matrix of one forms Thenonlinear equation to emerge is expressed as the vanishingof a traceless 3 times 3matrix of two forms Θ

Θ = 0 Θ = 119889Ω minus Ω and Ω (25)These can be considered as integrability conditions for (24)The closure property the gauge transformation and the gaugetheoretic interpretation can also hold for 3 times 3 systemsExplicitly the matrixΩ is given by

Ω = (

0 minus1205961

1205962

1205961

0 minus1205963

minus1205962

1205963

0

) (26)

Using (26) in (24) and (25) the following prolonged differen-tial system is obtained

1198891199101= 1205721minus 11991021205961+ 11991031205962 119889120596

1= 1205991minus 1205962and 1205963

1198891199102= 1205722+ 11991011205961minus 11991031205963 119889120596

2= 1205992minus 1205963and 1205961

1198891199103= 1205723minus 11991011205962+ 11991021205963 119889120596

3= 1205993minus 1205961and 1205962

(27)

By straightforward exterior differentiation of each 120572119894and

using (27) the followingTheorem results

Theorem 6 The 120572119894given in (27) have exterior derivatives

which are contained in the ring spanned by 1205991198943

1and 120572

1198953

1and

given explicitly as1198891205721= 11991021205991minus 11991031205992minus 1205961and 1205722+ 1205962and 1205723

1198891205722= minus 119910

11205991+ 11991031205993+ 1205961and 1205721minus 1205963and 1205723

1198891205723= 11991011205992minus 11991021205993minus 1205962and 1205721+ 1205963and 1205722

(28)

The one forms 1205721198943

1admit a series of Riccati representa-

tions which can be realized by defining six new one forms

1199102

11205724= 11991011205722minus 11991021205721 119910

2

11205725= 11991011205723minus 11991031205721

1199102

21205726= 11991021205721minus 11991011205722

1199102

21205727= 11991021205723minus 11991031205722 119910

2

31205728= 11991031205721minus 11991011205723

1199102

31205729= 11991031205722minus 11991021205723

(29)


In effect the larger 3times3 system is breaking up into several 2times2systems To write the new one forms explicitly introduce thenew functions 119910

1198959

4which are defined in terms of the original

1199101198943

1as follows

1199104=1199102

1199101

1199105=1199103

1199101

1199106=1199101

1199102

1199107=1199103

1199101

1199108=1199101

1199103

1199109=1199102

1199103

(30)

In terms of the functions defined in (30) the 1205721198959

4are given

as

1205724= 1198891199104minus (1 + 119910

2

4) 1205961+ 119910411991051205962+ 11991051205963

1205725= 1198891199105minus 119910411991051205961+ (1 + 119910

2

5) 1205962minus 11991041205963

1205726= 1198891199106+ (1 + 119910

2

6) 1205961minus 11991071205962minus 119910611991071205963

1205727= 1198891199107+ 119910611991071205961+ 11991061205962minus (1 + 119910

2

7) 1205963

1205728= 1198891199108+ 11991091205961minus (1 + 119910

2

8) 1205962+ 119910811991091205963

1205729= 1198891199109minus 11991081205961minus 119910811991091205962+ (1 + 119910

2

9) 1205963

(31)

To state the next theorem we need to define the followingthree matrices

Ω1= (

211991041205961minus 11991051205962

minus11991041205962minus 1205963

11991051205961+ 1205963

11991041205961minus 211991051205962

)

Ω2= (

minus211991061205961+ 11991071205963

1205962+ 11991061205963

minus11991071205961minus 1205966

minus11991061205961+ 211991071205963

)

Ω3= (

211991081205962minus 11991091205963

minus1205961minus 11991091205963

1205961+ 11991091205962

11991081205962minus 211991091205963

)

(32)

Theorem 7 The closure properties for the forms 1205721198949

4given in

(31) can be summarized in the form

119889(1205724

1205725

) = (minus (1 + 119910

2

4) 1205991+ 119910411991051205992+ 11991051205993

minus119910411991051205991+ (1 + 119910

2

5) 1205992minus 11991041205993

)

+ Ω1and (

1205724

1205725

)

119889 (1205726

1205727

) = ((1 + 119910

2

6) 1205991minus 11991071205992minus 119910611991071205993

119910611991071205991+ 11991061205992minus (1 + 119910

2

7) 1205993

)

+ Ω2and (

1205726

1205727

)

119889 (1205728

1205729

) = (11991011205991minus (1 + 119910

2

8) 1205992+ 119910811991091205993

minus11991081205991minus 119910811991091205992+ (1 + 119910

2

9) 1205993

)

+ Ω3and (

1205728

1205729

)

(33)

The exterior derivatives are therefore contained in the ringspanned by 120599

1198943

1and the 120572

1198959

4

In terms of Ω119894 a corresponding matrix of two forms Θ

119894

can be defined in terms of the corresponding Ω119894defined in

(32)

Θ119894= 119889Ω119894minus Ω119894and Ω119894 119894 = 1 2 3 (34)

Theorem 8 The 2 times 2matrix of two forms Θ119894defined by (34)

is contained in the ring of forms spanned by 1205991198943

1coupled with

either 1205724 1205725 1205726 1205727 or 120572

8 1205729when 119894 = 1 2 3 respectively

Proof DifferentiatingΩ1and simplifying we have

Θ1= 119889Ω1minus Ω1and Ω1

= (211991041205991minus 11991051205992+ 21205724and 1205961minus 1205725and 1205962

minus11991041205992minus 1205993minus 1205724and 1205962

11991051205991+ 1205993+ 1205725and 1205961

11991041205991minus 211991051205992+ 1205724and 1205961minus 21205725and 1205962

)

(35)

Similarmatrix expressions can be found for the cases inwhichΩ1is replaced by Ω

2orΩ3 respectively

Unlike Ω given in (26) the Ω119894in (32) are not traceless

Define their traces to be

120581119894= trΩ

119894 119894 = 1 2 3 (36)

These traces provide convenient ways of generating conser-vation laws on account of the followingTheorem

Theorem 9 The exterior derivatives of traces (36) are given by

1


1

31198891205812= minus 119910

61205991+ 11991071205993+ 1205961and 1205726minus 1205963and 1205727

1


(37)

Results (37) are contained in the ring spanned by 1205991198943

1and

1205721198959

4



To formulate a 3 times 3 119878119880(3) problem the set of Pfaffianequations

120572119894= 0 120572

119894= 119889119910119894minus Ω119894119895119910119895 119894 119895 = 1 2 3 (38)

are associated with the nonlinear equation In (38) Ω is atraceless 3 times 3 matrix consisting of a system of one formsThe nonlinear equation to be considered is expressed as thevanishing of a traceless 3times3matrix of two formsΘ exactly asin (25) which constitute the integrability condition for (38)

A 3times3matrix representation of the Lie algebra for 119878119880(3) isintroduced by means of generators 120582

119895for 119895 = 1 8 which

satisfy the following set of commutation relations

[120582119897 120582119898] = 2119894 119891

119897119898119899120582119899 (39)

The structure constants 119891119897119898119899

are totally antisymmetric in119897 119898 119899The one formΩ is expressed in terms of the generators120582119894as

Ω =

8

sum

119897=1

120596119897120582119897 (40)

Then the two forms Θ can be written as

Θ =

8

sum

119897=1

120599119897120582119897 120599119897= 119889120596119897minus 119894119891119897119898119899

120596119898and 120596119899 (41)

The nonlinear equation to be solved has the form 120599119897= 0 for

119897 = 1 8It will be useful to display (40) and (41) by using the

nonzero structure constants given in (39) since thesemay notbe readily accessible An explicit representation for thematrixΩ used here is given by

Ω = (

1205963+

1

radic3

1205968

1205961minus 1198941205962

1205964minus 1198941205965

1205961+ 1198941205962

minus1205963+

1

radic3

12059681205966minus 1198941205967

1205964+ 1198941205965

1205966+ 1198941205967

minus2

radic3

1205968

) (42)

Moreover in order that the presentation can be easier tofollow the eight forms 119889120596

119894specified by (41) will be given

explicitly

1198891205961= 1205991+ 21198941205962and 1205963+ 1198941205964and 1205967minus 1198941205965and 1205966

1198891205962= 1205992minus 21198941205961and 1205963+ 1198941205964and 1205966+ 1198941205965and 1205967

1198891205963= 1205993+ 21198941205961and 1205962+ 1198941205964and 1205965minus 1198941205966and 1205967

1198891205964= 1205994minus 1198941205961and 1205967minus 1198941205962and 1205966minus 1198941205963and 1205965+ radic3119894120596

5and 1205968

1198891205965= 1205995+ 1198941205961and 1205966minus 1198941205962and 1205967+ 1198941205963and 1205964minus radic3119894120596

4and 1205968

1198891205966= 1205996minus 1198941205961and 1205965+ 1198941205962and 1205964+ 1198941205963and 1205967+ radic3119894120596

7and 1205968

1198891205967= 1205997+ 1198941205961and 1205964+ 1198941205962and 1205965minus 1198941205963and 1205966minus radic3119894120596

6and 1205968

1198891205968= 1205998+ radic3119894120596

4and 1205965+ radic3119894120596

6and 1205967

(43)

Substituting (42) into (38) the following system of one forms1205721198943

1is obtained

1205721= 1198891199101minus (1205963+

1

radic3

1205968)1199101minus (1205961minus 1198941205962) 1199102minus (1205964minus 1198941205965) 1199103

1205722= 1198891199102minus (1205961+ 1198941205962) 1199101+ (1205963minus

1

radic3

1205968)1199102minus (1205966minus 1198941205967) 1199103

1205723= 1198891199103minus (1205964+ 1198941205965) 1199101minus (1205966+ 1198941205967) 1199102+

2

radic3

12059681199103

(44)

The increase in complexity of this system makes it oftennecessary to resort to the use of symbolic manipulation tocarry out longer calculations and for the most part onlyresults are given

Theorem10 Theexterior derivatives of the120572119894in (44) are given

by

1198891205721= minus 119910

2(1205991minus 1198941205992) minus 11991011205993minus 1199103(1205994minus 1198941205995)

minus1

radic3

11991011205998+ (1205963+

1

radic3

1205968) and 1205721

+ (1205961minus 1198941205962) and 1205722+ (1205964minus 1198941205965) and 1205723

1198891205722= minus 119910

1(1205991+ 1198941205992) + 11991021205993

minus1

radic3

11991021205998minus 1199103(1205996minus 1198941205997) + (120596

1+ 1198941205962) and 1205721

minus (1205963minus

1

radic3

1205968) and 1205722+ (1205966minus 1198941205967) and 1205723

1198891205723= minus 119910

1(1205994+ 1198941205995) minus 1199102(1205996+ 1198941205997)

+2

radic3

11991031205998+ (1205964+ 1198941205965) and 1205721

+ (1205966+ 1198941205967) and 1205722minus

2

radic3

1205968and 1205723

(45)

In order to keep the notation concise an abbreviation willbe introduced Suppose that 120574

119894 is a system of forms and 119894 119895

are integers then we make the abbreviation

120574119894119895plusmn

= 120574119894plusmn 119894120574119895 (46)

At this point quadratic pseudopotentials can be introducedin terms of the homogeneous variables of the same form as


(30) In the same way that (31) was produced the followingPfaffian equations based on the set 120572

1198943

1in (44) are obtained

1205724= 1198891199104minus 12059612+

+ 211991041205963+ 1199102

412059612minus

minus 119910512059667minus

+ 1199104119910512059645minus

1205725= 1198891199105minus 12059645+

+ 1199105(1205963+ radic3120596

8)

+ 1199102

512059645minus

minus 119910412059667+

+ 1199104119910512059612minus

1205726= 1198891199106minus 12059612minus

minus 211991061205963+ 1199102

612059612+

minus 119910712059645minus

+ 1199106119910712059667minus

1205727= 1198891199107minus 12059667+

minus 1199107(1205963minus radic3120596

8)

+ 1199102

712059667minus

minus 119910612059645+

+ 1199106119910712059612+

1205728= 1198891199108minus 12059645minus

minus 1199108(1205963+ radic3120596

8)

+ 1199102

812059645+

minus 119910912059612minus

+ 1199108119910912059667+

1205729= 1198891199109minus 12059667minus

+ 1199109(1205963minus radic3120596

8)

+ 1199102

912059667+

minus 119910812059612+

+ 1199108119910912059645+

(47)

These are coupledRiccati equations for the pairs of pseudopo-tentials (119910

4 1199105) (1199106 1199107) and (119910

8 1199109) respectively

Define the following set of 2 times 2matrices of one formsΩ119894

for 119894 = 1 2 3 as

Ω1=(minus21205963minus2119910412059612minusminus119910512059645minus 12059667minusminus119910412059645minus

12059667+minus119910512059612minus minus1205963minusradic31205968minus119910412059612minusminus2119910512059645minus

)

Ω2=(21205963minus2119910612059612+ minus 119910712059667minus 12059645minusminus119910612059667minus

12059645+minus119910712059612+ 1205963minusradic31205968minus119910612059612+minus2119910712059667minus

)

Ω3=(1205963+radic31205968minus2119910812059645+minus119910912059667+ 12059612minusminus119910812059667+

12059612+minus119910912059645+ minus1205963+radic31205968minus119910812059645+minus2119910912059667+

)

(48)

Making use of the matrices Ω119894defined in (48) the following

result can be stated

Theorem 11 The closure properties of the set of forms 1205721198949

4

given by (47) can be expressed in terms of the Ω119894as follows

119889(12057241205725)

= ((1199102

4minus 1) 1205991minus119894 (119910

2

4+ 1) 1205992+211991041205993+1199104119910512059945minusminus119910512059967minus

1199104119910512059912minus+11991051205993+(1199102

4minus 1) 1205994minus119894 (119910

2

5+ 1) 1205995minus119910412059967++

radic311991051205998

)

+Ω1 and (12057241205725)

119889 (12057261205727)

= ((1199102

6minus 1) 1205991+119894 (119910

2

6+ 1) 1205992minus211991061205993minus119910712059945minus+1199106119910712059967minus

1199106119910712059912+minus11991071205993minus119910612059945++(1199102

7minus 1) 1205996minus119894 (119910

2

7+ 1) 1205997+

radic311991071205998

)

+Ω2 and (12057261205727)

119889 (12057281205729)

= (minus119910912059912minusminus11991081205993+(119910

2

8minus 1) 1205994+119894 (119910

2

8+ 1) 1205995+1199108119910912059967+minus

radic311991081205998

minus119910812059912++11991091205993+1199108119910912059945++(1199102

9minus 1) 1205996+119894 (119910

2

9+ 1) 1205997minus

radic311991091205998

)

+Ω3 and (12057281205729)

(49)

Based on the forms Ω119894given in (48) we can differentiate

to define the following matrices of two forms 120585119894given by

120585119894= 119889Ω119894minus Ω119894and Ω119894 (50)

Theorem 12 The forms 120585119894defined in (50) are given explicitly

by the matrices

1205851

=(

212059612+ and 1205724+12059645minus and 1205725 12059645minus and 1205724minus119910412059945minus+12059967minus

minus2119910412059912minus minus 21205993 minus 119910512059945minus12059612minus and 1205724+212059645minus and 1205725

12059612minus and 1205725minus119910512059912minus+12059967+ minus119910412059912minusminus1205993 minus 212059945minus minusradic31205998

)

1205852

=(

212059612+ and 1205726+12059667minus and 1205727 12059667minus and 1205726+12059945minus minus 119910612059967minus

minus2119910612059912+ + 1205993 minus 119910712059967minus12059612+ and 1205726+212059667minus and 1205727minus119910612059912+

12059612+ and 1205727 minus 1199107 12059912+ + 12059945+ +1205993 minus 2119910712059967minus minusradic31205998

)

1205853

=(

212059645+ and 1205728+12059667+ and 1205729+1205993 12059667+ and 1205728+12059912minusminus119910812059967+

minus2119910812059945+ minus 119910912059967+ +radic3120599812059645+ and 1205728+212059667+ and 1205729minus1205993

12059645+ and 1205729+12059912+minus119910912059945+ minus119910812059945+ minus 2119910912059967+ +radic31205998

)

(51)

From these results it is concluded that 1205851is contained in

the ring of 1205724 1205725 and the 120599

119897 1205852is contained in the ring of

1205726 1205727 and 120599

119897 and 120585

3is in the ring of 120572

8 1205729 and 120599

119897

The two forms 1205851198943

1therefore vanish for the solutions of the

nonlinear equations (39) and of the coupled Riccati equations120572119895= 09

4 As in the case of 119874(3) the one forms Ω

119894are not

traceless Denoting the trace of Ω119894by

120591119894= trΩ

119894 (52)

the following theorem then follows

Theorem 13 The closure properties of the exterior derivativesof the traces 120591

119894are given by

1

31198891205911= 12059612minus

and 1205724+ 12059645minus

and 1205725

minus 119910412059912minus

minus 1205993minus 119910512059945minus

minus1

radic3

1205998


1

31198891205912= 12059612+

and 1205726+ 12059667minus

and 1205727

minus 119910612059912+

+ 1205993minus 119910712059967minus

minus1

radic3

1205998

1

31198891205913= 12059645+

and 1205728+ 12059667+

and 1205729

minus 119910812059945+

minus 119910912059967+

+2

radic3

1205998

(53)

The one forms (52) then can be used to generate con-servation laws Note that as in the 119874(3) case a 3 times 3

problemΩ has been reduced to three separate 2times2 problemsin terms of the matrices Ω

119897 Further prolongation can be

continued beginning with the one formsΩ119897given in (48) To

briefly outline the further steps in the procedure a system ofPfaffians

119897119895 is introduced which are based on the one forms

Ω119897

119897119895= 0

119897119895= 119889119910119897119895minus (Ω119897)119895119896119910119897119896 (54)

The subscript 119897 in 119910119897119895and

119897119895indicates that they belong to the

subsystem defined byΩ119897 and it is not summed over Exterior

differentiation of 119897119895yields

119889119897119895= (Ω119897)119895119896and 119897119896minus (Θ)

119895119896119910119897119896 (55)

The 119910119897119895 are pseudopotentials for the original nonlinear

equation (39) Next quadratic pseudopotentials are includedas the homogeneous variables and the prolongation can becontinued

5 A Conservation Law and Summary

These types of results turn out to be very useful for furtherstudy of integrable equationsThey can be used for generatinginfinite numbers of conservation laws In addition the resultsare independent of any further structure of the forms 120596

119894in

all cases In fact the one form Ω need not be thought of asuniqueThis is due to the fact thatΩ andΘ are form invariantunder the gauge transformationΩ rarr 120596

1015840= 119889119860 119860

minus1+119860Ω119860

minus1

and Θ rarr Θ1015840= 119860Θ119860

minus1 where in the 119878119871(2R) case 119860 is anarbitrary 2 times 2 space-time-dependent matrix of determinantone The gauge transformation property holds in the 3 times 3

case as well The pseudopotentials serve as potentials forconservation laws in a generalized senseThey can be definedunder a choice of Pfaffian forms such as

120572119894= 0 120572

119894= 119889119910119894+ 119865119894119889119909 + 119866

119894119889119905 (56)

with the property that the exterior derivatives 119889120572119894are con-

tained in the ring spanned by 120572119894 and 120599

119897

119889120572119894= sum

119895

119860119894119895and 120572119895+sum

119897

Γ119894119897120599119897 (57)

This can be thought of as a generalization of the FrobeniusTheorem for complete integrability of Pfaffian systems

Theorem 3 implies that 119889120590119894= 0when 119894 = 1 2 for solutions

of the original equation If either of these forms is expressedas 120590 = I119889119909 +J119889119905 then 119889120590 = 0 implies that

120597I

120597119905minus120597J

120597119905= 0 (58)

ThusI is a conserved density andJ a conserved current Letus finally represent 120596

1= 1198861119889119909 + 119887

1119889119905 1205962= 1198862119889119909 + 119887

2119889119905 and

1205963= 120578119889119909 + 119887

3119889119905 where 120578 is a parameter The 119909-dependent

piece of 1205723in (17) implies

1199103119909

+ 211988611199103+ 11988621199102

3minus 120578 = 0 (59)

By substituting an asymptotic expansion around 120578 = infin for1199103in (59) of the form

1199103=

infin

sum

0

120578minus119899119884119899 (60)

a recursion for the 119884119899is obtained Thus (59) becomes

sum

119899

120578minus119899119884119899119909

+ 21198861sum

119899

120578minus119899119884119899

+ 1198862(sum

119899

120578minus119899119884119899)

2

minus 120578 = 0

(61)

Expanding the power collecting coefficients of 120578 and equat-ing coefficients of 120578 to zero we get

1198841119909

+ 211988611198841= 1 119884

119899119909+ 21198861119884119899+ 1198862

119899minus1

sum

119896=1

119884119899minus119896

119884119896= 0

(62)

The consistency of solving 1205723by using just the 119909 part is

guaranteed by complete integrability The 119909 part of the form1205901is expressed as

(1205901)119909= (1198861+

infin

sum

119899=1

120578minus1198991198841198991198862)119889119909 (63)

Consequently from (58) the 119899th conserved density is

I119899= 1198862119884119899 (64)

The existence of links between the types of prolongationhere and other types of prolongations which are based onclosed differential systems is a subject for further work

References

[1] F B Estabrook and H D Wahlquist ldquoClassical geometriesdefined by exterior differential systems on higher frame bun-dlesrdquo Classical and Quantum Gravity vol 6 no 3 pp 263ndash2741989

[2] H Cartan Differential Forms Dover Mineola NY USA 2006[3] M J Ablowitz D J Kaup A C Newell and H Segur ldquoMethod

for solving the sine-Gordon equationrdquo Physical Review Lettersvol 30 pp 1262ndash1264 1973


[4] M J Ablowitz D K Kaup A C Newell and H SegurldquoNonlinear-evolution equations of physical significancerdquo Phys-ical Review Letters vol 31 pp 125ndash127 1973

[5] R Sasaki ldquoPseudopotentials for the general AKNS systemrdquoPhysics Letters A vol 73 no 2 pp 77ndash80 1979

[6] R Sasaki ldquoGeometric approach to soliton equationsrdquo Proceed-ings of the Royal Society London vol 373 no 1754 pp 373ndash3841980

[7] M Crampin ldquoSolitons and 119878119871(2R)rdquo Physics Letters A vol 66no 3 pp 170ndash172 1978

[8] D J Kaup and J Yang ldquoThe inverse scattering transform andsquared eigenfunctions for a degenerate 3 times 3 operatorrdquo InverseProblems vol 25 no 10 Article ID 105010 2009

[9] D J Kaup and R A Van Gorder ldquoThe inverse scatteringtransform and squared eigenfunctions for the nondegenerate3 times 3 operator and its soliton structurerdquo Inverse Problems vol26 no 5 Article ID 055005 2010

[10] P Bracken ldquoIntrinsic formulation of geometric integrabilityand associated Riccati system generating conservation lawsrdquoInternational Journal of Geometric Methods in Modern Physicsvol 6 no 5 pp 825ndash837 2009

[11] D J Kaup and R A Van Gorder ldquoSquared eigenfunctions andthe perturbation theory for the nondegenerate N timesN operatora general outlinerdquo Journal of Physics A vol 43 Article ID434019 2010

[12] H D Wahlquist and F B Estabrook ldquoProlongation structuresof nonlinear evolution equationsrdquo Journal of MathematicalPhysics vol 16 pp 1ndash7 1975

[13] P Bracken ldquoOn two-dimensional manifolds with constantGaussian curvature and their associated equationsrdquo Interna-tional Journal of Geometric Methods in Modern Physics vol 9no 3 Article ID 1250018 2012

Submit your manuscripts athttpwwwhindawicom

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


the end a collection of conservation laws are developed fromone of the results and some speculation as to how this typerelates to other procedures [12 13]

2 119878119871(2R) Prolongations

The procedure begins by associating a system of Pfaffianequations to the nonlinear system to be studied namely

120572119894= 0 120572

119894= 119889119910119894minus Ω119894119895119910119895 119894 119895 = 1 2 (1)

In (1) Ω is a traceless 2 times 2 matrix which consists of afamily of one forms 120596

119894 At the end it is desired to express

these one forms in terms of independent variables which arecalled 119909 and 119905 the dependent variables and their derivativesHowever no specific equation need be assumed at the startThe underlying structure group is 119878119871(2R) Explicitly thematrix of one forms is given by the 2 times 2matrix

Ω = (1205961

1205962

1205963minus1205961

) (2)

The integrability conditions are expressed as the vanishing ofa traceless 2 times 2matrix of two forms Θ given by

Θ = 119889Ω minus Ω and Ω (3)

In terms of components the left-hand side of (3) has the form

Θ = (Θ119894119895) = (

1205991

1205992

1205993minus1205991

) (4)

Substituting (2) into (3) a matrix which contains the integra-bility equations in terms of the basis one forms is obtainedThe three equations are

1205991= 1198891205961minus 1205962and 1205963 120599

2= 1198891205962minus 21205961and 1205962

1205993= 1198891205963+ 21205961and 1205963

(5)

Of use in the theorems which follow it is also useful to havethe 119889120596

119894in (5) expressed explicitly in terms of the 120599

119894as

1198891205961= 1205991+ 1205962and 1205963 119889120596

2= 1205992+ 21205961and 1205962

1198891205963= 1205993minus 21205961and 1205963

(6)

Therefore by selecting a particular choice for the set 120596119894 the

nonlinear equation of interest can be written simply as

Θ = 0 120599119894= 0 119894 = 1 2 3 (7)

It is easy to see that the system is closed Upon differentiatingΘ and substituting (3) we obtain

119889Θ = Ω and Θ minus Θ and Ω (8)

This implies that the exterior derivatives of the set of 120599119894 are

contained in the ring of two forms 120599119894

Now the differential ideal can be prolonged by includingthe forms 120572

1 1205722 given by (1)

Theorem 1 The differential one forms

1205721= 1198891199101minus 12059611199101minus 12059621199102 120572

2= 1198891199102minus 12059631199101+ 12059611199102 (9)

have the following exterior derivatives

1198891205721= 1205961and 1205721+ 1205962and 1205722minus 11991011205991minus 11991021205992

1198891205722= minus 120596

1and 1205722+ 1205963and 1205721+ 11991021205991minus 11991011205993

(10)

Hence the exterior derivatives of the 120572119894 are contained in the

ring of forms 120599119894 120572119895

Proof The forms (9) follow by substituting (2) into (1)Differentiating 120572

1in (9) it is found that

1198891205721= minus11991011198891205961+ 1205961and 1198891199101minus 11991021198891205962+ 1205962and 1198891199102 (11)

Obtaining 1198891199101and 119889119910

2from (9) and 119889120596

1 1198891205962from (6) 119889120572

1

becomes

1198891205721= minus11991011205991minus 11991011205962and 1205963+ 1205961and 1205721+ 11991021205961and 1205962minus 11991021205992

minus 211991021205961and 1205962+ 1205962and 1205722+ 11991011205962and 1205963minus 11991021205962and 1205961

= 1205961and 1205721+ 1205962and 1205722minus 11991011205991minus 11991021205992

(12)

Similarly beginning with 1205722

1198891205722= minus11991011198891205963+ 1205963and 1198891199101+ 11991021198891205961minus 1205961and 1198891199102

= minus11991011205993+ 211991011205961and 1205963+ 1205963and 1205721

+ 11991011205963and 1205961+ 11991021205963and 1205962

+ 11991021205991+ 11991021205962and 1205963minus 1205961and 1205722minus 11991011205961and 1205963

= minus1205961and 1205722+ 1205963and 1205721+ 11991021205991minus 11991011205993

(13)

as required

Corollary 2 The exterior derivatives of the 120572119894in (9) can be

expressed concisely in terms of the matrix elements ofΘ andΩas

119889120572119894= minusΘ119894119895119910119895+ Ω119894119895and 120572119895 119894 119895 = 1 2 (14)

The corollary is Theorem 1 after using the definitions ofthe matrices in (2) and (4)

The forms 1205721 1205722given in (9) lead to a natural Riccati

representation called 1205723 1205724for this differential system This

arises by taking the following linear combinations of 1205721and

1205722

1199102

11205723= 11991011205722minus 11991021205721

= 11991011198891199102minus 11991021198891199101minus 1199102

11205963+ 2119910111991021205961+ 1199102

21205962

1199102

21205724= 11991021205721minus 11991011205722

= 11991021198891199101minus 11991011198891199102minus 1199102

21205962minus 2119910111991021205961+ 1199102

11205963

(15)



4can be


1199103=1199102

1199101

1199104=1199101

1199102

(16)


1205723= 1198891199103minus 1205963+ 211991031205961+ 1199102

31205962

1205724= 1198891199104minus 1205962minus 211991041205961+ 1199102

41205963

(17)


4


derivatives

1198891205723= 211991031205991+ 1199102

31205992minus 1205993+ 21205723and (1205961+ 11991031205962)

1198891205724= minus 2119910

41205991minus 1205992+ 1199102

41205993+ 21205724and (minus120596

1+ 11991041205963)

(18)


119895


2= minus120596

1+


1and 120590

2

are found to be

1198891205901= 1205991+ 11991031205992+ 1205723and 1205962

1198891205902= minus1205991+ 11991031205992+ 1205724and 1205962

(19)


2equiv 0 mod 120599

119894 120572119895


3 1205724


5and 119910



5and 120572




6as

1205725= 1198891199105minus 1205961minus 11991031205962 120572

6= 1198891199106+ 1205961minus 11991041205963 (20)



6= 1205991minus 11991041205993minus 1205724and 1205963

(21)









8by means of

1205727= 1198891199107minus 119890minus211991051205962 120572

8= 1198891199108minus 119890minus211991061205963 (22)


8are given by



(23)






120572119894= 0 120572

119894= 119889119910119894minus Ω119894119895119910119895 119894 119895 = 1 2 3 (24)



Ω = (

0 minus1205961

1205962

1205961

0 minus1205963

minus1205962

1205963

0

) (26)


1198891199101= 1205721minus 11991021205961+ 11991031205962 119889120596

1= 1205991minus 1205962and 1205963

1198891199102= 1205722+ 11991011205961minus 11991031205963 119889120596

2= 1205992minus 1205963and 1205961

1198891199103= 1205723minus 11991011205962+ 11991021205963 119889120596

3= 1205993minus 1205961and 1205962

(27)





1and 120572

1198953

1and


1198891205722= minus 119910

11205991+ 11991031205993+ 1205961and 1205721minus 1205963and 1205723


(28)




1199102

11205724= 11991011205722minus 11991021205721 119910

2

11205725= 11991011205723minus 11991031205721

1199102

21205726= 11991021205721minus 11991011205722

1199102

21205727= 11991021205723minus 11991031205722 119910

2

31205728= 11991031205721minus 11991011205723

1199102

31205729= 11991031205722minus 11991021205723

(29)



1198959


1199101198943

1as follows

1199104=1199102

1199101

1199105=1199103

1199101

1199106=1199101

1199102

1199107=1199103

1199101

1199108=1199101

1199103

1199109=1199102

1199103

(30)


4are given

as

1205724= 1198891199104minus (1 + 119910

2

4) 1205961+ 119910411991051205962+ 11991051205963

1205725= 1198891199105minus 119910411991051205961+ (1 + 119910

2

5) 1205962minus 11991041205963

1205726= 1198891199106+ (1 + 119910

2

6) 1205961minus 11991071205962minus 119910611991071205963

1205727= 1198891199107+ 119910611991071205961+ 11991061205962minus (1 + 119910

2

7) 1205963

1205728= 1198891199108+ 11991091205961minus (1 + 119910

2

8) 1205962+ 119910811991091205963

1205729= 1198891199109minus 11991081205961minus 119910811991091205962+ (1 + 119910

2

9) 1205963

(31)


Ω1= (

211991041205961minus 11991051205962

minus11991041205962minus 1205963

11991051205961+ 1205963

11991041205961minus 211991051205962

)

Ω2= (

minus211991061205961+ 11991071205963

1205962+ 11991061205963

minus11991071205961minus 1205966

minus11991061205961+ 211991071205963

)

Ω3= (

211991081205962minus 11991091205963

minus1205961minus 11991091205963

1205961+ 11991091205962

11991081205962minus 211991091205963

)

(32)


4given in


119889(1205724

1205725

) = (minus (1 + 119910

2

4) 1205991+ 119910411991051205992+ 11991051205993

minus119910411991051205991+ (1 + 119910

2

5) 1205992minus 11991041205993

)

+ Ω1and (

1205724

1205725

)

119889 (1205726

1205727

) = ((1 + 119910

2

6) 1205991minus 11991071205992minus 119910611991071205993

119910611991071205991+ 11991061205992minus (1 + 119910

2

7) 1205993

)

+ Ω2and (

1205726

1205727

)

119889 (1205728

1205729

) = (11991011205991minus (1 + 119910

2

8) 1205992+ 119910811991091205993

minus11991081205991minus 119910811991091205992+ (1 + 119910

2

9) 1205993

)

+ Ω3and (

1205728

1205729

)

(33)


1198943

1and the 120572

1198959

4


119894


(32)

Θ119894= 119889Ω119894minus Ω119894and Ω119894 119894 = 1 2 3 (34)



1coupled with

either 1205724 1205725 1205726 1205727 or 120572






11991051205991+ 1205993+ 1205725and 1205961


)

(35)


2orΩ3 respectively



120581119894= trΩ

119894 119894 = 1 2 3 (36)



1


1

31198891205812= minus 119910

61205991+ 11991071205993+ 1205961and 1205726minus 1205963and 1205727

1


(37)


1and

1205721198959

4




120572119894= 0 120572

119894= 119889119910119894minus Ω119894119895119910119895 119894 119895 = 1 2 3 (38)



119895for 119895 = 1 8 which


[120582119897 120582119898] = 2119894 119891

119897119898119899120582119899 (39)



Ω =

8

sum

119897=1

120596119897120582119897 (40)


Θ =

8

sum

119897=1

120599119897120582119897 120599119897= 119889120596119897minus 119894119891119897119898119899

120596119898and 120596119899 (41)




Ω = (

1205963+

1

radic3

1205968

1205961minus 1198941205962

1205964minus 1198941205965

1205961+ 1198941205962

minus1205963+

1

radic3

12059681205966minus 1198941205967

1205964+ 1198941205965

1205966+ 1198941205967

minus2

radic3

1205968

) (42)



explicitly

1198891205961= 1205991+ 21198941205962and 1205963+ 1198941205964and 1205967minus 1198941205965and 1205966

1198891205962= 1205992minus 21198941205961and 1205963+ 1198941205964and 1205966+ 1198941205965and 1205967

1198891205963= 1205993+ 21198941205961and 1205962+ 1198941205964and 1205965minus 1198941205966and 1205967


5and 1205968


4and 1205968


7and 1205968


6and 1205968

1198891205968= 1205998+ radic3119894120596

4and 1205965+ radic3119894120596

6and 1205967

(43)


1is obtained

1205721= 1198891199101minus (1205963+

1

radic3


1205722= 1198891199102minus (1205961+ 1198941205962) 1199101+ (1205963minus

1

radic3

1205968)1199102minus (1205966minus 1198941205967) 1199103

1205723= 1198891199103minus (1205964+ 1198941205965) 1199101minus (1205966+ 1198941205967) 1199102+

2

radic3

12059681199103

(44)



by

1198891205721= minus 119910


minus1

radic3

11991011205998+ (1205963+

1

radic3

1205968) and 1205721


1198891205722= minus 119910

1(1205991+ 1198941205992) + 11991021205993

minus1

radic3

11991021205998minus 1199103(1205996minus 1198941205997) + (120596

1+ 1198941205962) and 1205721

minus (1205963minus

1

radic3

1205968) and 1205722+ (1205966minus 1198941205967) and 1205723

1198891205723= minus 119910

1(1205994+ 1198941205995) minus 1199102(1205996+ 1198941205997)

+2

radic3

11991031205998+ (1205964+ 1198941205965) and 1205721

+ (1205966+ 1198941205967) and 1205722minus

2

radic3

1205968and 1205723

(45)




120574119894119895plusmn

= 120574119894plusmn 119894120574119895 (46)




1198943


1205724= 1198891199104minus 12059612+

+ 211991041205963+ 1199102

412059612minus

minus 119910512059667minus

+ 1199104119910512059645minus

1205725= 1198891199105minus 12059645+

+ 1199105(1205963+ radic3120596

8)

+ 1199102

512059645minus

minus 119910412059667+

+ 1199104119910512059612minus

1205726= 1198891199106minus 12059612minus

minus 211991061205963+ 1199102

612059612+

minus 119910712059645minus

+ 1199106119910712059667minus

1205727= 1198891199107minus 12059667+


8)

+ 1199102

712059667minus

minus 119910612059645+

+ 1199106119910712059612+

1205728= 1198891199108minus 12059645minus

minus 1199108(1205963+ radic3120596

8)

+ 1199102

812059645+

minus 119910912059612minus

+ 1199108119910912059667+

1205729= 1198891199109minus 12059667minus

+ 1199109(1205963minus radic3120596

8)

+ 1199102

912059667+

minus 119910812059612+

+ 1199108119910912059645+

(47)


4 1199105) (1199106 1199107) and (119910



for 119894 = 1 2 3 as



)



)



)

(48)




4


119889(12057241205725)

= ((1199102

4minus 1) 1205991minus119894 (119910

2


1199104119910512059912minus+11991051205993+(1199102

4minus 1) 1205994minus119894 (119910

2

5+ 1) 1205995minus119910412059967++

radic311991051205998

)

+Ω1 and (12057241205725)

119889 (12057261205727)

= ((1199102

6minus 1) 1205991+119894 (119910

2


1199106119910712059912+minus11991071205993minus119910612059945++(1199102

7minus 1) 1205996minus119894 (119910

2

7+ 1) 1205997+

radic311991071205998

)

+Ω2 and (12057261205727)

119889 (12057281205729)


2

8minus 1) 1205994+119894 (119910

2

8+ 1) 1205995+1199108119910912059967+minus

radic311991081205998

minus119910812059912++11991091205993+1199108119910912059945++(1199102

9minus 1) 1205996+119894 (119910

2

9+ 1) 1205997minus

radic311991091205998

)

+Ω3 and (12057281205729)

(49)



120585119894= 119889Ω119894minus Ω119894and Ω119894 (50)


by the matrices

1205851

=(




)

1205852

=(




)

1205853

=(




)

(51)




1205726 1205727 and 120599

119897 and 120585


8 1205729 and 120599

119897





119894are not


120591119894= trΩ

119894 (52)



119894are given by

1

31198891205911= 12059612minus

and 1205724+ 12059645minus

and 1205725

minus 119910412059912minus


minus1

radic3

1205998


1

31198891205912= 12059612+

and 1205726+ 12059667minus

and 1205727

minus 119910612059912+

+ 1205993minus 119910712059967minus

minus1

radic3

1205998

1

31198891205913= 12059645+

and 1205728+ 12059667+

and 1205729

minus 119910812059945+

minus 119910912059967+

+2

radic3

1205998

(53)







Ω119897

119897119895= 0

119897119895= 119889119910119897119895minus (Ω119897)119895119896119910119897119896 (54)





119889119897119895= (Ω119897)119895119896and 119897119896minus (Θ)

119895119896119910119897119896 (55)





119894in


1015840= 119889119860 119860

minus1+119860Ω119860

minus1

and Θ rarr Θ1015840= 119860Θ119860



120572119894= 0 120572

119894= 119889119910119894+ 119865119894119889119909 + 119866

119894119889119905 (56)



119897

119889120572119894= sum

119895

119860119894119895and 120572119895+sum

119897

Γ119894119897120599119897 (57)




120597I

120597119905minus120597J

120597119905= 0 (58)


1= 1198861119889119909 + 119887

1119889119905 1205962= 1198862119889119909 + 119887

2119889119905 and

1205963= 120578119889119909 + 119887



1199103119909

+ 211988611199103+ 11988621199102

3minus 120578 = 0 (59)


1199103=

infin

sum

0

120578minus119899119884119899 (60)


sum

119899

120578minus119899119884119899119909

+ 21198861sum

119899

120578minus119899119884119899

+ 1198862(sum

119899

120578minus119899119884119899)

2

minus 120578 = 0

(61)


1198841119909

+ 211988611198841= 1 119884

119899119909+ 21198861119884119899+ 1198862

119899minus1

sum

119896=1

119884119899minus119896

119884119896= 0

(62)



(1205901)119909= (1198861+

infin

sum

119899=1

120578minus1198991198841198991198862)119889119909 (63)


I119899= 1198862119884119899 (64)


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











4can be


1199103=1199102

1199101

1199104=1199101

1199102

(16)


1205723= 1198891199103minus 1205963+ 211991031205961+ 1199102

31205962

1205724= 1198891199104minus 1205962minus 211991041205961+ 1199102

41205963

(17)


4


derivatives

1198891205723= 211991031205991+ 1199102

31205992minus 1205993+ 21205723and (1205961+ 11991031205962)

1198891205724= minus 2119910

41205991minus 1205992+ 1199102

41205993+ 21205724and (minus120596

1+ 11991041205963)

(18)


119895


2= minus120596

1+


1and 120590

2

are found to be

1198891205901= 1205991+ 11991031205992+ 1205723and 1205962

1198891205902= minus1205991+ 11991031205992+ 1205724and 1205962

(19)


2equiv 0 mod 120599

119894 120572119895


3 1205724


5and 119910



5and 120572




6as

1205725= 1198891199105minus 1205961minus 11991031205962 120572

6= 1198891199106+ 1205961minus 11991041205963 (20)



6= 1205991minus 11991041205993minus 1205724and 1205963

(21)









8by means of

1205727= 1198891199107minus 119890minus211991051205962 120572

8= 1198891199108minus 119890minus211991061205963 (22)


8are given by



(23)






120572119894= 0 120572

119894= 119889119910119894minus Ω119894119895119910119895 119894 119895 = 1 2 3 (24)



Ω = (

0 minus1205961

1205962

1205961

0 minus1205963

minus1205962

1205963

0

) (26)


1198891199101= 1205721minus 11991021205961+ 11991031205962 119889120596

1= 1205991minus 1205962and 1205963

1198891199102= 1205722+ 11991011205961minus 11991031205963 119889120596

2= 1205992minus 1205963and 1205961

1198891199103= 1205723minus 11991011205962+ 11991021205963 119889120596

3= 1205993minus 1205961and 1205962

(27)





1and 120572

1198953

1and


1198891205722= minus 119910

11205991+ 11991031205993+ 1205961and 1205721minus 1205963and 1205723


(28)




1199102

11205724= 11991011205722minus 11991021205721 119910

2

11205725= 11991011205723minus 11991031205721

1199102

21205726= 11991021205721minus 11991011205722

1199102

21205727= 11991021205723minus 11991031205722 119910

2

31205728= 11991031205721minus 11991011205723

1199102

31205729= 11991031205722minus 11991021205723

(29)



1198959


1199101198943

1as follows

1199104=1199102

1199101

1199105=1199103

1199101

1199106=1199101

1199102

1199107=1199103

1199101

1199108=1199101

1199103

1199109=1199102

1199103

(30)


4are given

as

1205724= 1198891199104minus (1 + 119910

2

4) 1205961+ 119910411991051205962+ 11991051205963

1205725= 1198891199105minus 119910411991051205961+ (1 + 119910

2

5) 1205962minus 11991041205963

1205726= 1198891199106+ (1 + 119910

2

6) 1205961minus 11991071205962minus 119910611991071205963

1205727= 1198891199107+ 119910611991071205961+ 11991061205962minus (1 + 119910

2

7) 1205963

1205728= 1198891199108+ 11991091205961minus (1 + 119910

2

8) 1205962+ 119910811991091205963

1205729= 1198891199109minus 11991081205961minus 119910811991091205962+ (1 + 119910

2

9) 1205963

(31)


Ω1= (

211991041205961minus 11991051205962

minus11991041205962minus 1205963

11991051205961+ 1205963

11991041205961minus 211991051205962

)

Ω2= (

minus211991061205961+ 11991071205963

1205962+ 11991061205963

minus11991071205961minus 1205966

minus11991061205961+ 211991071205963

)

Ω3= (

211991081205962minus 11991091205963

minus1205961minus 11991091205963

1205961+ 11991091205962

11991081205962minus 211991091205963

)

(32)


4given in


119889(1205724

1205725

) = (minus (1 + 119910

2

4) 1205991+ 119910411991051205992+ 11991051205993

minus119910411991051205991+ (1 + 119910

2

5) 1205992minus 11991041205993

)

+ Ω1and (

1205724

1205725

)

119889 (1205726

1205727

) = ((1 + 119910

2

6) 1205991minus 11991071205992minus 119910611991071205993

119910611991071205991+ 11991061205992minus (1 + 119910

2

7) 1205993

)

+ Ω2and (

1205726

1205727

)

119889 (1205728

1205729

) = (11991011205991minus (1 + 119910

2

8) 1205992+ 119910811991091205993

minus11991081205991minus 119910811991091205992+ (1 + 119910

2

9) 1205993

)

+ Ω3and (

1205728

1205729

)

(33)


1198943

1and the 120572

1198959

4


119894


(32)

Θ119894= 119889Ω119894minus Ω119894and Ω119894 119894 = 1 2 3 (34)



1coupled with

either 1205724 1205725 1205726 1205727 or 120572






11991051205991+ 1205993+ 1205725and 1205961


)

(35)


2orΩ3 respectively



120581119894= trΩ

119894 119894 = 1 2 3 (36)



1


1

31198891205812= minus 119910

61205991+ 11991071205993+ 1205961and 1205726minus 1205963and 1205727

1


(37)


1and

1205721198959

4




120572119894= 0 120572

119894= 119889119910119894minus Ω119894119895119910119895 119894 119895 = 1 2 3 (38)



119895for 119895 = 1 8 which


[120582119897 120582119898] = 2119894 119891

119897119898119899120582119899 (39)



Ω =

8

sum

119897=1

120596119897120582119897 (40)


Θ =

8

sum

119897=1

120599119897120582119897 120599119897= 119889120596119897minus 119894119891119897119898119899

120596119898and 120596119899 (41)




Ω = (

1205963+

1

radic3

1205968

1205961minus 1198941205962

1205964minus 1198941205965

1205961+ 1198941205962

minus1205963+

1

radic3

12059681205966minus 1198941205967

1205964+ 1198941205965

1205966+ 1198941205967

minus2

radic3

1205968

) (42)



explicitly

1198891205961= 1205991+ 21198941205962and 1205963+ 1198941205964and 1205967minus 1198941205965and 1205966

1198891205962= 1205992minus 21198941205961and 1205963+ 1198941205964and 1205966+ 1198941205965and 1205967

1198891205963= 1205993+ 21198941205961and 1205962+ 1198941205964and 1205965minus 1198941205966and 1205967


5and 1205968


4and 1205968


7and 1205968


6and 1205968

1198891205968= 1205998+ radic3119894120596

4and 1205965+ radic3119894120596

6and 1205967

(43)


1is obtained

1205721= 1198891199101minus (1205963+

1

radic3


1205722= 1198891199102minus (1205961+ 1198941205962) 1199101+ (1205963minus

1

radic3

1205968)1199102minus (1205966minus 1198941205967) 1199103

1205723= 1198891199103minus (1205964+ 1198941205965) 1199101minus (1205966+ 1198941205967) 1199102+

2

radic3

12059681199103

(44)



by

1198891205721= minus 119910


minus1

radic3

11991011205998+ (1205963+

1

radic3

1205968) and 1205721


1198891205722= minus 119910

1(1205991+ 1198941205992) + 11991021205993

minus1

radic3

11991021205998minus 1199103(1205996minus 1198941205997) + (120596

1+ 1198941205962) and 1205721

minus (1205963minus

1

radic3

1205968) and 1205722+ (1205966minus 1198941205967) and 1205723

1198891205723= minus 119910

1(1205994+ 1198941205995) minus 1199102(1205996+ 1198941205997)

+2

radic3

11991031205998+ (1205964+ 1198941205965) and 1205721

+ (1205966+ 1198941205967) and 1205722minus

2

radic3

1205968and 1205723

(45)




120574119894119895plusmn

= 120574119894plusmn 119894120574119895 (46)




1198943


1205724= 1198891199104minus 12059612+

+ 211991041205963+ 1199102

412059612minus

minus 119910512059667minus

+ 1199104119910512059645minus

1205725= 1198891199105minus 12059645+

+ 1199105(1205963+ radic3120596

8)

+ 1199102

512059645minus

minus 119910412059667+

+ 1199104119910512059612minus

1205726= 1198891199106minus 12059612minus

minus 211991061205963+ 1199102

612059612+

minus 119910712059645minus

+ 1199106119910712059667minus

1205727= 1198891199107minus 12059667+


8)

+ 1199102

712059667minus

minus 119910612059645+

+ 1199106119910712059612+

1205728= 1198891199108minus 12059645minus

minus 1199108(1205963+ radic3120596

8)

+ 1199102

812059645+

minus 119910912059612minus

+ 1199108119910912059667+

1205729= 1198891199109minus 12059667minus

+ 1199109(1205963minus radic3120596

8)

+ 1199102

912059667+

minus 119910812059612+

+ 1199108119910912059645+

(47)


4 1199105) (1199106 1199107) and (119910



for 119894 = 1 2 3 as



)



)



)

(48)




4


119889(12057241205725)

= ((1199102

4minus 1) 1205991minus119894 (119910

2


1199104119910512059912minus+11991051205993+(1199102

4minus 1) 1205994minus119894 (119910

2

5+ 1) 1205995minus119910412059967++

radic311991051205998

)

+Ω1 and (12057241205725)

119889 (12057261205727)

= ((1199102

6minus 1) 1205991+119894 (119910

2


1199106119910712059912+minus11991071205993minus119910612059945++(1199102

7minus 1) 1205996minus119894 (119910

2

7+ 1) 1205997+

radic311991071205998

)

+Ω2 and (12057261205727)

119889 (12057281205729)


2

8minus 1) 1205994+119894 (119910

2

8+ 1) 1205995+1199108119910912059967+minus

radic311991081205998

minus119910812059912++11991091205993+1199108119910912059945++(1199102

9minus 1) 1205996+119894 (119910

2

9+ 1) 1205997minus

radic311991091205998

)

+Ω3 and (12057281205729)

(49)



120585119894= 119889Ω119894minus Ω119894and Ω119894 (50)


by the matrices

1205851

=(




)

1205852

=(




)

1205853

=(




)

(51)




1205726 1205727 and 120599

119897 and 120585


8 1205729 and 120599

119897





119894are not


120591119894= trΩ

119894 (52)



119894are given by

1

31198891205911= 12059612minus

and 1205724+ 12059645minus

and 1205725

minus 119910412059912minus


minus1

radic3

1205998


1

31198891205912= 12059612+

and 1205726+ 12059667minus

and 1205727

minus 119910612059912+

+ 1205993minus 119910712059967minus

minus1

radic3

1205998

1

31198891205913= 12059645+

and 1205728+ 12059667+

and 1205729

minus 119910812059945+

minus 119910912059967+

+2

radic3

1205998

(53)







Ω119897

119897119895= 0

119897119895= 119889119910119897119895minus (Ω119897)119895119896119910119897119896 (54)





119889119897119895= (Ω119897)119895119896and 119897119896minus (Θ)

119895119896119910119897119896 (55)





119894in


1015840= 119889119860 119860

minus1+119860Ω119860

minus1

and Θ rarr Θ1015840= 119860Θ119860



120572119894= 0 120572

119894= 119889119910119894+ 119865119894119889119909 + 119866

119894119889119905 (56)



119897

119889120572119894= sum

119895

119860119894119895and 120572119895+sum

119897

Γ119894119897120599119897 (57)




120597I

120597119905minus120597J

120597119905= 0 (58)


1= 1198861119889119909 + 119887

1119889119905 1205962= 1198862119889119909 + 119887

2119889119905 and

1205963= 120578119889119909 + 119887



1199103119909

+ 211988611199103+ 11988621199102

3minus 120578 = 0 (59)


1199103=

infin

sum

0

120578minus119899119884119899 (60)


sum

119899

120578minus119899119884119899119909

+ 21198861sum

119899

120578minus119899119884119899

+ 1198862(sum

119899

120578minus119899119884119899)

2

minus 120578 = 0

(61)


1198841119909

+ 211988611198841= 1 119884

119899119909+ 21198861119884119899+ 1198862

119899minus1

sum

119896=1

119884119899minus119896

119884119896= 0

(62)



(1205901)119909= (1198861+

infin

sum

119899=1

120578minus1198991198841198991198862)119889119909 (63)


I119899= 1198862119884119899 (64)


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198959


1199101198943

1as follows

1199104=1199102

1199101

1199105=1199103

1199101

1199106=1199101

1199102

1199107=1199103

1199101

1199108=1199101

1199103

1199109=1199102

1199103

(30)


4are given

as

1205724= 1198891199104minus (1 + 119910

2

4) 1205961+ 119910411991051205962+ 11991051205963

1205725= 1198891199105minus 119910411991051205961+ (1 + 119910

2

5) 1205962minus 11991041205963

1205726= 1198891199106+ (1 + 119910

2

6) 1205961minus 11991071205962minus 119910611991071205963

1205727= 1198891199107+ 119910611991071205961+ 11991061205962minus (1 + 119910

2

7) 1205963

1205728= 1198891199108+ 11991091205961minus (1 + 119910

2

8) 1205962+ 119910811991091205963

1205729= 1198891199109minus 11991081205961minus 119910811991091205962+ (1 + 119910

2

9) 1205963

(31)


Ω1= (

211991041205961minus 11991051205962

minus11991041205962minus 1205963

11991051205961+ 1205963

11991041205961minus 211991051205962

)

Ω2= (

minus211991061205961+ 11991071205963

1205962+ 11991061205963

minus11991071205961minus 1205966

minus11991061205961+ 211991071205963

)

Ω3= (

211991081205962minus 11991091205963

minus1205961minus 11991091205963

1205961+ 11991091205962

11991081205962minus 211991091205963

)

(32)


4given in


119889(1205724

1205725

) = (minus (1 + 119910

2

4) 1205991+ 119910411991051205992+ 11991051205993

minus119910411991051205991+ (1 + 119910

2

5) 1205992minus 11991041205993

)

+ Ω1and (

1205724

1205725

)

119889 (1205726

1205727

) = ((1 + 119910

2

6) 1205991minus 11991071205992minus 119910611991071205993

119910611991071205991+ 11991061205992minus (1 + 119910

2

7) 1205993

)

+ Ω2and (

1205726

1205727

)

119889 (1205728

1205729

) = (11991011205991minus (1 + 119910

2

8) 1205992+ 119910811991091205993

minus11991081205991minus 119910811991091205992+ (1 + 119910

2

9) 1205993

)

+ Ω3and (

1205728

1205729

)

(33)


1198943

1and the 120572

1198959

4


119894


(32)

Θ119894= 119889Ω119894minus Ω119894and Ω119894 119894 = 1 2 3 (34)



1coupled with

either 1205724 1205725 1205726 1205727 or 120572






11991051205991+ 1205993+ 1205725and 1205961


)

(35)


2orΩ3 respectively



120581119894= trΩ

119894 119894 = 1 2 3 (36)



1


1

31198891205812= minus 119910

61205991+ 11991071205993+ 1205961and 1205726minus 1205963and 1205727

1


(37)


1and

1205721198959

4




120572119894= 0 120572

119894= 119889119910119894minus Ω119894119895119910119895 119894 119895 = 1 2 3 (38)



119895for 119895 = 1 8 which


[120582119897 120582119898] = 2119894 119891

119897119898119899120582119899 (39)



Ω =

8

sum

119897=1

120596119897120582119897 (40)


Θ =

8

sum

119897=1

120599119897120582119897 120599119897= 119889120596119897minus 119894119891119897119898119899

120596119898and 120596119899 (41)




Ω = (

1205963+

1

radic3

1205968

1205961minus 1198941205962

1205964minus 1198941205965

1205961+ 1198941205962

minus1205963+

1

radic3

12059681205966minus 1198941205967

1205964+ 1198941205965

1205966+ 1198941205967

minus2

radic3

1205968

) (42)



explicitly

1198891205961= 1205991+ 21198941205962and 1205963+ 1198941205964and 1205967minus 1198941205965and 1205966

1198891205962= 1205992minus 21198941205961and 1205963+ 1198941205964and 1205966+ 1198941205965and 1205967

1198891205963= 1205993+ 21198941205961and 1205962+ 1198941205964and 1205965minus 1198941205966and 1205967


5and 1205968


4and 1205968


7and 1205968


6and 1205968

1198891205968= 1205998+ radic3119894120596

4and 1205965+ radic3119894120596

6and 1205967

(43)


1is obtained

1205721= 1198891199101minus (1205963+

1

radic3


1205722= 1198891199102minus (1205961+ 1198941205962) 1199101+ (1205963minus

1

radic3

1205968)1199102minus (1205966minus 1198941205967) 1199103

1205723= 1198891199103minus (1205964+ 1198941205965) 1199101minus (1205966+ 1198941205967) 1199102+

2

radic3

12059681199103

(44)



by

1198891205721= minus 119910


minus1

radic3

11991011205998+ (1205963+

1

radic3

1205968) and 1205721


1198891205722= minus 119910

1(1205991+ 1198941205992) + 11991021205993

minus1

radic3

11991021205998minus 1199103(1205996minus 1198941205997) + (120596

1+ 1198941205962) and 1205721

minus (1205963minus

1

radic3

1205968) and 1205722+ (1205966minus 1198941205967) and 1205723

1198891205723= minus 119910

1(1205994+ 1198941205995) minus 1199102(1205996+ 1198941205997)

+2

radic3

11991031205998+ (1205964+ 1198941205965) and 1205721

+ (1205966+ 1198941205967) and 1205722minus

2

radic3

1205968and 1205723

(45)




120574119894119895plusmn

= 120574119894plusmn 119894120574119895 (46)




1198943


1205724= 1198891199104minus 12059612+

+ 211991041205963+ 1199102

412059612minus

minus 119910512059667minus

+ 1199104119910512059645minus

1205725= 1198891199105minus 12059645+

+ 1199105(1205963+ radic3120596

8)

+ 1199102

512059645minus

minus 119910412059667+

+ 1199104119910512059612minus

1205726= 1198891199106minus 12059612minus

minus 211991061205963+ 1199102

612059612+

minus 119910712059645minus

+ 1199106119910712059667minus

1205727= 1198891199107minus 12059667+


8)

+ 1199102

712059667minus

minus 119910612059645+

+ 1199106119910712059612+

1205728= 1198891199108minus 12059645minus

minus 1199108(1205963+ radic3120596

8)

+ 1199102

812059645+

minus 119910912059612minus

+ 1199108119910912059667+

1205729= 1198891199109minus 12059667minus

+ 1199109(1205963minus radic3120596

8)

+ 1199102

912059667+

minus 119910812059612+

+ 1199108119910912059645+

(47)


4 1199105) (1199106 1199107) and (119910



for 119894 = 1 2 3 as



)



)



)

(48)




4


119889(12057241205725)

= ((1199102

4minus 1) 1205991minus119894 (119910

2


1199104119910512059912minus+11991051205993+(1199102

4minus 1) 1205994minus119894 (119910

2

5+ 1) 1205995minus119910412059967++

radic311991051205998

)

+Ω1 and (12057241205725)

119889 (12057261205727)

= ((1199102

6minus 1) 1205991+119894 (119910

2


1199106119910712059912+minus11991071205993minus119910612059945++(1199102

7minus 1) 1205996minus119894 (119910

2

7+ 1) 1205997+

radic311991071205998

)

+Ω2 and (12057261205727)

119889 (12057281205729)


2

8minus 1) 1205994+119894 (119910

2

8+ 1) 1205995+1199108119910912059967+minus

radic311991081205998

minus119910812059912++11991091205993+1199108119910912059945++(1199102

9minus 1) 1205996+119894 (119910

2

9+ 1) 1205997minus

radic311991091205998

)

+Ω3 and (12057281205729)

(49)



120585119894= 119889Ω119894minus Ω119894and Ω119894 (50)


by the matrices

1205851

=(




)

1205852

=(




)

1205853

=(




)

(51)




1205726 1205727 and 120599

119897 and 120585


8 1205729 and 120599

119897





119894are not


120591119894= trΩ

119894 (52)



119894are given by

1

31198891205911= 12059612minus

and 1205724+ 12059645minus

and 1205725

minus 119910412059912minus


minus1

radic3

1205998


1

31198891205912= 12059612+

and 1205726+ 12059667minus

and 1205727

minus 119910612059912+

+ 1205993minus 119910712059967minus

minus1

radic3

1205998

1

31198891205913= 12059645+

and 1205728+ 12059667+

and 1205729

minus 119910812059945+

minus 119910912059967+

+2

radic3

1205998

(53)







Ω119897

119897119895= 0

119897119895= 119889119910119897119895minus (Ω119897)119895119896119910119897119896 (54)





119889119897119895= (Ω119897)119895119896and 119897119896minus (Θ)

119895119896119910119897119896 (55)





119894in


1015840= 119889119860 119860

minus1+119860Ω119860

minus1

and Θ rarr Θ1015840= 119860Θ119860



120572119894= 0 120572

119894= 119889119910119894+ 119865119894119889119909 + 119866

119894119889119905 (56)



119897

119889120572119894= sum

119895

119860119894119895and 120572119895+sum

119897

Γ119894119897120599119897 (57)




120597I

120597119905minus120597J

120597119905= 0 (58)


1= 1198861119889119909 + 119887

1119889119905 1205962= 1198862119889119909 + 119887

2119889119905 and

1205963= 120578119889119909 + 119887



1199103119909

+ 211988611199103+ 11988621199102

3minus 120578 = 0 (59)


1199103=

infin

sum

0

120578minus119899119884119899 (60)


sum

119899

120578minus119899119884119899119909

+ 21198861sum

119899

120578minus119899119884119899

+ 1198862(sum

119899

120578minus119899119884119899)

2

minus 120578 = 0

(61)


1198841119909

+ 211988611198841= 1 119884

119899119909+ 21198861119884119899+ 1198862

119899minus1

sum

119896=1

119884119899minus119896

119884119896= 0

(62)



(1205901)119909= (1198861+

infin

sum

119899=1

120578minus1198991198841198991198862)119889119909 (63)


I119899= 1198862119884119899 (64)


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra












120572119894= 0 120572

119894= 119889119910119894minus Ω119894119895119910119895 119894 119895 = 1 2 3 (38)



119895for 119895 = 1 8 which


[120582119897 120582119898] = 2119894 119891

119897119898119899120582119899 (39)



Ω =

8

sum

119897=1

120596119897120582119897 (40)


Θ =

8

sum

119897=1

120599119897120582119897 120599119897= 119889120596119897minus 119894119891119897119898119899

120596119898and 120596119899 (41)




Ω = (

1205963+

1

radic3

1205968

1205961minus 1198941205962

1205964minus 1198941205965

1205961+ 1198941205962

minus1205963+

1

radic3

12059681205966minus 1198941205967

1205964+ 1198941205965

1205966+ 1198941205967

minus2

radic3

1205968

) (42)



explicitly

1198891205961= 1205991+ 21198941205962and 1205963+ 1198941205964and 1205967minus 1198941205965and 1205966

1198891205962= 1205992minus 21198941205961and 1205963+ 1198941205964and 1205966+ 1198941205965and 1205967

1198891205963= 1205993+ 21198941205961and 1205962+ 1198941205964and 1205965minus 1198941205966and 1205967


5and 1205968


4and 1205968


7and 1205968


6and 1205968

1198891205968= 1205998+ radic3119894120596

4and 1205965+ radic3119894120596

6and 1205967

(43)


1is obtained

1205721= 1198891199101minus (1205963+

1

radic3


1205722= 1198891199102minus (1205961+ 1198941205962) 1199101+ (1205963minus

1

radic3

1205968)1199102minus (1205966minus 1198941205967) 1199103

1205723= 1198891199103minus (1205964+ 1198941205965) 1199101minus (1205966+ 1198941205967) 1199102+

2

radic3

12059681199103

(44)



by

1198891205721= minus 119910


minus1

radic3

11991011205998+ (1205963+

1

radic3

1205968) and 1205721


1198891205722= minus 119910

1(1205991+ 1198941205992) + 11991021205993

minus1

radic3

11991021205998minus 1199103(1205996minus 1198941205997) + (120596

1+ 1198941205962) and 1205721

minus (1205963minus

1

radic3

1205968) and 1205722+ (1205966minus 1198941205967) and 1205723

1198891205723= minus 119910

1(1205994+ 1198941205995) minus 1199102(1205996+ 1198941205997)

+2

radic3

11991031205998+ (1205964+ 1198941205965) and 1205721

+ (1205966+ 1198941205967) and 1205722minus

2

radic3

1205968and 1205723

(45)




120574119894119895plusmn

= 120574119894plusmn 119894120574119895 (46)




1198943


1205724= 1198891199104minus 12059612+

+ 211991041205963+ 1199102

412059612minus

minus 119910512059667minus

+ 1199104119910512059645minus

1205725= 1198891199105minus 12059645+

+ 1199105(1205963+ radic3120596

8)

+ 1199102

512059645minus

minus 119910412059667+

+ 1199104119910512059612minus

1205726= 1198891199106minus 12059612minus

minus 211991061205963+ 1199102

612059612+

minus 119910712059645minus

+ 1199106119910712059667minus

1205727= 1198891199107minus 12059667+


8)

+ 1199102

712059667minus

minus 119910612059645+

+ 1199106119910712059612+

1205728= 1198891199108minus 12059645minus

minus 1199108(1205963+ radic3120596

8)

+ 1199102

812059645+

minus 119910912059612minus

+ 1199108119910912059667+

1205729= 1198891199109minus 12059667minus

+ 1199109(1205963minus radic3120596

8)

+ 1199102

912059667+

minus 119910812059612+

+ 1199108119910912059645+

(47)


4 1199105) (1199106 1199107) and (119910



for 119894 = 1 2 3 as



)



)



)

(48)




4


119889(12057241205725)

= ((1199102

4minus 1) 1205991minus119894 (119910

2


1199104119910512059912minus+11991051205993+(1199102

4minus 1) 1205994minus119894 (119910

2

5+ 1) 1205995minus119910412059967++

radic311991051205998

)

+Ω1 and (12057241205725)

119889 (12057261205727)

= ((1199102

6minus 1) 1205991+119894 (119910

2


1199106119910712059912+minus11991071205993minus119910612059945++(1199102

7minus 1) 1205996minus119894 (119910

2

7+ 1) 1205997+

radic311991071205998

)

+Ω2 and (12057261205727)

119889 (12057281205729)


2

8minus 1) 1205994+119894 (119910

2

8+ 1) 1205995+1199108119910912059967+minus

radic311991081205998

minus119910812059912++11991091205993+1199108119910912059945++(1199102

9minus 1) 1205996+119894 (119910

2

9+ 1) 1205997minus

radic311991091205998

)

+Ω3 and (12057281205729)

(49)



120585119894= 119889Ω119894minus Ω119894and Ω119894 (50)


by the matrices

1205851

=(




)

1205852

=(




)

1205853

=(




)

(51)




1205726 1205727 and 120599

119897 and 120585


8 1205729 and 120599

119897





119894are not


120591119894= trΩ

119894 (52)



119894are given by

1

31198891205911= 12059612minus

and 1205724+ 12059645minus

and 1205725

minus 119910412059912minus


minus1

radic3

1205998


1

31198891205912= 12059612+

and 1205726+ 12059667minus

and 1205727

minus 119910612059912+

+ 1205993minus 119910712059967minus

minus1

radic3

1205998

1

31198891205913= 12059645+

and 1205728+ 12059667+

and 1205729

minus 119910812059945+

minus 119910912059967+

+2

radic3

1205998

(53)







Ω119897

119897119895= 0

119897119895= 119889119910119897119895minus (Ω119897)119895119896119910119897119896 (54)





119889119897119895= (Ω119897)119895119896and 119897119896minus (Θ)

119895119896119910119897119896 (55)





119894in


1015840= 119889119860 119860

minus1+119860Ω119860

minus1

and Θ rarr Θ1015840= 119860Θ119860



120572119894= 0 120572

119894= 119889119910119894+ 119865119894119889119909 + 119866

119894119889119905 (56)



119897

119889120572119894= sum

119895

119860119894119895and 120572119895+sum

119897

Γ119894119897120599119897 (57)




120597I

120597119905minus120597J

120597119905= 0 (58)


1= 1198861119889119909 + 119887

1119889119905 1205962= 1198862119889119909 + 119887

2119889119905 and

1205963= 120578119889119909 + 119887



1199103119909

+ 211988611199103+ 11988621199102

3minus 120578 = 0 (59)


1199103=

infin

sum

0

120578minus119899119884119899 (60)


sum

119899

120578minus119899119884119899119909

+ 21198861sum

119899

120578minus119899119884119899

+ 1198862(sum

119899

120578minus119899119884119899)

2

minus 120578 = 0

(61)


1198841119909

+ 211988611198841= 1 119884

119899119909+ 21198861119884119899+ 1198862

119899minus1

sum

119896=1

119884119899minus119896

119884119896= 0

(62)



(1205901)119909= (1198861+

infin

sum

119899=1

120578minus1198991198841198991198862)119889119909 (63)


I119899= 1198862119884119899 (64)


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198943


1205724= 1198891199104minus 12059612+

+ 211991041205963+ 1199102

412059612minus

minus 119910512059667minus

+ 1199104119910512059645minus

1205725= 1198891199105minus 12059645+

+ 1199105(1205963+ radic3120596

8)

+ 1199102

512059645minus

minus 119910412059667+

+ 1199104119910512059612minus

1205726= 1198891199106minus 12059612minus

minus 211991061205963+ 1199102

612059612+

minus 119910712059645minus

+ 1199106119910712059667minus

1205727= 1198891199107minus 12059667+


8)

+ 1199102

712059667minus

minus 119910612059645+

+ 1199106119910712059612+

1205728= 1198891199108minus 12059645minus

minus 1199108(1205963+ radic3120596

8)

+ 1199102

812059645+

minus 119910912059612minus

+ 1199108119910912059667+

1205729= 1198891199109minus 12059667minus

+ 1199109(1205963minus radic3120596

8)

+ 1199102

912059667+

minus 119910812059612+

+ 1199108119910912059645+

(47)


4 1199105) (1199106 1199107) and (119910



for 119894 = 1 2 3 as



)



)



)

(48)




4


119889(12057241205725)

= ((1199102

4minus 1) 1205991minus119894 (119910

2


1199104119910512059912minus+11991051205993+(1199102

4minus 1) 1205994minus119894 (119910

2

5+ 1) 1205995minus119910412059967++

radic311991051205998

)

+Ω1 and (12057241205725)

119889 (12057261205727)

= ((1199102

6minus 1) 1205991+119894 (119910

2


1199106119910712059912+minus11991071205993minus119910612059945++(1199102

7minus 1) 1205996minus119894 (119910

2

7+ 1) 1205997+

radic311991071205998

)

+Ω2 and (12057261205727)

119889 (12057281205729)


2

8minus 1) 1205994+119894 (119910

2

8+ 1) 1205995+1199108119910912059967+minus

radic311991081205998

minus119910812059912++11991091205993+1199108119910912059945++(1199102

9minus 1) 1205996+119894 (119910

2

9+ 1) 1205997minus

radic311991091205998

)

+Ω3 and (12057281205729)

(49)



120585119894= 119889Ω119894minus Ω119894and Ω119894 (50)


by the matrices

1205851

=(




)

1205852

=(




)

1205853

=(




)

(51)




1205726 1205727 and 120599

119897 and 120585


8 1205729 and 120599

119897





119894are not


120591119894= trΩ

119894 (52)



119894are given by

1

31198891205911= 12059612minus

and 1205724+ 12059645minus

and 1205725

minus 119910412059912minus


minus1

radic3

1205998


1

31198891205912= 12059612+

and 1205726+ 12059667minus

and 1205727

minus 119910612059912+

+ 1205993minus 119910712059967minus

minus1

radic3

1205998

1

31198891205913= 12059645+

and 1205728+ 12059667+

and 1205729

minus 119910812059945+

minus 119910912059967+

+2

radic3

1205998

(53)







Ω119897

119897119895= 0

119897119895= 119889119910119897119895minus (Ω119897)119895119896119910119897119896 (54)





119889119897119895= (Ω119897)119895119896and 119897119896minus (Θ)

119895119896119910119897119896 (55)





119894in


1015840= 119889119860 119860

minus1+119860Ω119860

minus1

and Θ rarr Θ1015840= 119860Θ119860



120572119894= 0 120572

119894= 119889119910119894+ 119865119894119889119909 + 119866

119894119889119905 (56)



119897

119889120572119894= sum

119895

119860119894119895and 120572119895+sum

119897

Γ119894119897120599119897 (57)




120597I

120597119905minus120597J

120597119905= 0 (58)


1= 1198861119889119909 + 119887

1119889119905 1205962= 1198862119889119909 + 119887

2119889119905 and

1205963= 120578119889119909 + 119887



1199103119909

+ 211988611199103+ 11988621199102

3minus 120578 = 0 (59)


1199103=

infin

sum

0

120578minus119899119884119899 (60)


sum

119899

120578minus119899119884119899119909

+ 21198861sum

119899

120578minus119899119884119899

+ 1198862(sum

119899

120578minus119899119884119899)

2

minus 120578 = 0

(61)


1198841119909

+ 211988611198841= 1 119884

119899119909+ 21198861119884119899+ 1198862

119899minus1

sum

119896=1

119884119899minus119896

119884119896= 0

(62)



(1205901)119909= (1198861+

infin

sum

119899=1

120578minus1198991198841198991198862)119889119909 (63)


I119899= 1198862119884119899 (64)


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

31198891205912= 12059612+

and 1205726+ 12059667minus

and 1205727

minus 119910612059912+

+ 1205993minus 119910712059967minus

minus1

radic3

1205998

1

31198891205913= 12059645+

and 1205728+ 12059667+

and 1205729

minus 119910812059945+

minus 119910912059967+

+2

radic3

1205998

(53)







Ω119897

119897119895= 0

119897119895= 119889119910119897119895minus (Ω119897)119895119896119910119897119896 (54)





119889119897119895= (Ω119897)119895119896and 119897119896minus (Θ)

119895119896119910119897119896 (55)





119894in


1015840= 119889119860 119860

minus1+119860Ω119860

minus1

and Θ rarr Θ1015840= 119860Θ119860



120572119894= 0 120572

119894= 119889119910119894+ 119865119894119889119909 + 119866

119894119889119905 (56)



119897

119889120572119894= sum

119895

119860119894119895and 120572119895+sum

119897

Γ119894119897120599119897 (57)




120597I

120597119905minus120597J

120597119905= 0 (58)


1= 1198861119889119909 + 119887

1119889119905 1205962= 1198862119889119909 + 119887

2119889119905 and

1205963= 120578119889119909 + 119887



1199103119909

+ 211988611199103+ 11988621199102

3minus 120578 = 0 (59)


1199103=

infin

sum

0

120578minus119899119884119899 (60)


sum

119899

120578minus119899119884119899119909

+ 21198861sum

119899

120578minus119899119884119899

+ 1198862(sum

119899

120578minus119899119884119899)

2

minus 120578 = 0

(61)


1198841119909

+ 211988611198841= 1 119884

119899119909+ 21198861119884119899+ 1198862

119899minus1

sum

119896=1

119884119899minus119896

119884119896= 0

(62)



(1205901)119909= (1198861+

infin

sum

119899=1

120578minus1198991198841198991198862)119889119909 (63)


I119899= 1198862119884119899 (64)


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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