Research ArticleOn the Conservation Laws and Exact Solutions ofa Modified Hunter-Saxton Equation

Sait San1 and Emrullah YaGar2

1 Department of Mathematics-Computer Art-Science Faculty Eskisehir Osmangazi University 26480 Eskisehir Turkey2Department of Mathematics Faculty of Arts and Sciences Uludag University 16059 Bursa Turkey

Correspondence should be addressed to Emrullah Yasar eyasaruludagedutr

Received 30 January 2014 Accepted 24 March 2014 Published 10 April 2014

Academic Editor Fabien Gatti

Copyright copy 2014 S San and E Yasar This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We study the modified Hunter-Saxton equation which arises in modelling of nematic liquid crystals We obtain local conservationlaws using the nonlocal conservation method and multiplier approach In addition using the relationship between conservationlaws and Lie-point symmetries some reductions and exact solutions are obtained

1 Introduction

It is well known that in order to obtain the physical meaningsof the equation considered below conservation laws are thekey instruments They can be observed in a variety of fieldssuch as obtaining the numerical schemas Lyapunov stabilityanalysis and numerical integration In the literature thereexist a lot ofmethods (see [1ndash7]) A detailed reviewof existingmethods in the literature can be found in [8] In additionwe observe some valuable software computer packages in thisarea [9 10]

In this work we study the modified Hunter-Saxton(MHS) equation

119906119905minus 2119906119909119906119909119909minus 119906119909119909119909

119906 minus 119906119909119909119905

= 0 (1)

which is a third order nonlinear partial differential equation(PDE) This equation has been first suggested by Hunterand Saxton [11] for the theoretical modeling of nematicliquid crystals They showed that the weakly nonlinear wavesare described by (1) where 119906(119909 119905) describes the directorfield of a nematic liquid crystal 119909 is a space variable in areference framemovingwith the linearizedwave velocity and119905 is a slow time variable [11 12] Geometric interpretationsand integrability properties of (1) are studied by someauthors [13 14] Johnpillai and Khalique [12] showed thatthe underlying equation admits three parameter Lie-point

symmetry generators Using these generators they obtainedan optimal system of one-dimensional subalgebras Symme-try reductions and exact solutions are obtained Moreoverusing the variational method they constructed an infinitenumber of nonlocal conservation laws by the transformationof the dependent variable of the underlying equation In [15]Nadjafikhah and Ahangari investigated the Lie symmetriesand conservation laws of second order nonlinear hyperbolicHunter-Saxton equation (HSE) The conservation laws of theHSE are computed via three different methods includingBoyerrsquos generalization of Noetherrsquos theorem first homotopymethod and second homotopy method

In this work we investigate local conservation laws of (1)For this aim we consider Ibragimovrsquos nonlocal conservationand Steudelrsquos multiplier methods respectively In additionwe obtain some reductions and exact solutions using therelationship between conservation laws and Lie-point sym-metries [16]

The outline of the paper is as follows In Section 2 wediscuss some main operator identities and their relationshipThen in Section 3 we briefly give nonlocal conservationmultiplier and double reduction methods In Section 4local symmetry generators are constructed with two distinctmethods In this section symmetry reductions and exactsolutions are also obtained Finally in Section 5 conclusionsare presented

Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2014 Article ID 349059 6 pageshttpdxdoiorg1011552014349059

2 Advances in Mathematical Physics

2 Preliminaries

We briefly present notation to be used and recall basic defini-tions and theorems which utilize below [2 7 16] Considerthe 119896th-order system of PDEs of 119899 independent variables119909 = (119909

1 1199092 119909

119899) and 119898 dependent variables 119906 = (119906

1

1199062 119906

119898)

119864120572(119909 119906 119906

(1) 119906

(119896)) = 0 120572 = 1 119898 (2)

where 119906(119894)

is the collection of 119894th-order partial derivatives119906120572

119894= 119863119894(119906120572) 119906120572119894119895= 119863119895119863119894(119906120572) respectively with the total

differentiation operator with respect to 119909119894 given by

119863119894=

120597

120597119909119894+ 119906120572

119894

120597

120597119906120572+ 119906120572

119894119895

120597

120597119906120572

119895

+ sdot sdot sdot 119894 = 1 119899 (3)

in which the summation convention is used The Lie-pointgenerator is

119883 = 120585119894 120597

120597119909119894

+ 120578120572 120597

120597119906120572 (4)

where 120585119894 and 120578

120572 are functions of only independent anddependent functionsThe operator (4) is an abbreviated formof the infinite formal sum

119883 = 120585119894 120597

120597119909119894

+ 120578120572 120597

120597119906120572+sum

119904ge1

120577120572

11989411198942sdotsdotsdot119894119904

120597

120597119906120572

11989411198942sdotsdotsdot119894119904

(5)

where the additional coefficients can be determined from theprolongation formulae

120577120572

119894= 119863119894(120578120572) minus 120585119895119906120572

119895119894

120577120572

1198941sdotsdotsdot119894119904

= 1198631198941

sdot sdot sdot 119863119894119904

(120577120572

1198941sdotsdotsdot119894119904minus1

) minus 120585119895119906120572

1198951198941sdotsdotsdot119894119904

119904 gt 1

(6)

The Noether operators associated with a Lie-point generator119883 are

119873119894= 120585119894+119882120572 120575

120575119906120572

119894

+

infin

sum

119904ge1

1198631198941

sdot sdot sdot 119863119894119904

120597

120597119906120572

1198941sdotsdotsdot119894119904

119894 = 1 2 119899

(7)

in which119882120572 is the Lie characteristic function

119882120572= 120578120572minus 120585119895119906120572

119895 (8)

The conserved vector of (2) where each119879119894 isin 119860119860 is the spaceof all differential functions satisfies the equation

119863119894119879119894

|(2)= 0 (9)

along the solution of (2)

3 Conservation Laws Methods

31 Nonlocal Conservation Method We will denote indepen-dent variables119909 = (119909

1 1199092)with1199091 = 1199091199092 = 119905 one dependent

variable 119906 together with its derivatives up to119901 arbitrary orderThe 119901th-order PDE

119864 (119909 119906 1199061 119906

119901) = 0 (10)

has always formal Lagrangian Formal Lagrangian is mul-tiplication of a new adjoint variable 119908(119909 119905) with a givenequation Namely

119871 = 119908120572119864120572 (11)

With this formal Lagrangian

119864lowast=

120575119871

120575119906120572

(12)

adjoint equation is constructed Here 120575120575119906 is the Euler-Lagrange operator and defined by

120575

120575119906120572=

120597

120597119906120572+

infin

sum

119904ge1

(minus1)1199041198631198941

sdot sdot sdot 119863119894119904

120597

120597119906120572

1198941sdotsdotsdot119894119904

120572 = 1 119898

(13)

Theorem 1 (see [7]) Every Lie-point Lie-Backlund and non-local symmetry of (2) gives a conservation law for the equationunder consideration The conserved vector components aredetermined with

119879119894= 120585119894119871 +119882

120572[

120597119871

120597119906119894

minus 119863119895(

120597119871

120597119906119894119895

) + 119863119895119863119896(

120597119871

120597119906119894119895119896

)

minus119863119895119863119896119863119898(

120597119871

120597119906119894119895119896119898

)]

+ 119863119895(119882120572) [

120597119871

120597119906119894119895

minus 119863119896(

120597119871

120597119906119894119895119896

) + 119863119896119863119898(

120597119871

120597119906119894119895119896119898

)]

+ 119863119895119863119896(119882120572) [

120597119871

120597119906119894119895119896

minus 119863119898(

120597119871

120597119906119894119895119896119898

)]

+ 119863119895119863119896119863119898(119882120572) [(

120597119871

120597119906119894119895119896119898

)]

(14)

where Lagrangian (formal Lagrangian) function is given by

119871 = 119908120572119864120572(119909 119906 119906

(1) 119906

(119896)) (15)

120585119894 120578120572 are the coefficient functions of the associated generator(4)

The conserved vectors obtained from (14) involve thearbitrary solutions 119908 of the adjoint equation (12) and henceone obtains an infinite number of conservation laws for (1) bychoosing 119908

Definition 2 We say that (2) is strictly self-adjoint if theadjoint equation (12) becomes equivalent to (2) after the sub-stitution 119908 = 119906

120575119871

120575119906120572= 120582119864 (119909 119905 119906 119906

119909 119906

119909119909119905) (16)

with 120582 being generic coefficient

Advances in Mathematical Physics 3

Definition 3 We say that (2) is quasi-self-adjoint if the adjointequation (12) becomes equivalent to (2) after the substitution119908 = 120601(119906) 120601(119906) = 0

32 The Multiplier Method Amultiplier Λ120572(119909 119906 119906

119909 ) has

the property that

Λ120572119864120572= 119863119894119879119894 (17)

holds identically Here we will consider multipliers of thirdorder that is Λ

120572= Λ

120572(119909 119905 119906 119906

119909 119906119909119909 119906119909119909119909

) The righthand side of (17) is a divergence expressionThe determiningequation for the multiplier Λ

120572is

120575 (Λ120572119864120572)

120575119906120572

= 0 (18)

Once the multipliers are obtained the conserved vectors arecalculated via a homotopy formula [5 17] All the multiplierscan be calculated with the aid of (18) for which the equationcan be expressed as a local conservation law [9]

33 Double Reduction Method Let 119883 be any Lie-point sym-metry and 119879119894 are the components of conserved vector If 119883and 119879 satisfy

119883(119879119894) + 119879119894119863119895(120585119895) minus 119879119895119863119895(120585119894) = 0 119894 = 1 2 (19)

then 119883 is associated with 119879 We define a nonlocal variable Vby 119879119905 = V

119909 119879119909 = minusV

119905 Taking the similarity variables 119903 119904 120603

with the generator119883 = 120597120597119904 we have in similarity variables

119879119903= V119904 119879

119904= minusV119903 (20)

so that the conservation law is rewritten as

119863119903119879119903+ 119863119904119879119904= 0 (21)

Using the chain rule we have

119863119909= 119863119909(119904)119863119904+ 119863119909(119903)119863119903 119863

119905= 119863119905(119904)119863119904+ 119863119905(119903)119863119903

(22)

so that

V119909= V119904119863119909(119904) + V

119903119863119909(119903) V

119905= V119904119863119905(119904) + V

119903119863119905(119903) (23)

and so

119879119905= 119879119903119863119909(119904) minus 119879

119904119863119909(119903) 119879

119909= 119879119903119863119909(119904) minus 119879

119904119863119905(119903)

(24)

Using the above linear algebraical system we can get

119879119904=

119879119905119863119905(119904) + 119879

119909119863119909(119904)

119863119905(119903)119863119909(119904) minus 119863

119909(119903)119863119905(119904)

(25)

119879119903=

119879119905119863119905(119903) + 119879

119909119863119909(119903)

119863119905(119903)119863119909(119904) minus 119863

119909(119903)119863119905(119904)

(26)

The components 119879119909 119879119905 depend on (119905 119909 119906 119906(1) 119906(2)

119906(119902minus1)

) which means that 119879119904 119879119903 depend on (119904 119903 120603 120603

119903

120603119903119903 120603

119903(119902minus1)) for solutions invariant under 119883 Therefore

(21) becomes (120597119879119904120597119904) + 119863119903119879119903= 0

For 119879 associated with 119883 we have 119883119879119903 = 0 and 119883119879119904 = 0Thus 119879119903 and 119879119903 are invariant under 119883 This means 120597119879119904120597119904 =0 and 120597119879

119903120597119904 = 0 so that 119879119903(119903 120603 120603

119903 120603119903119903 120603

119903(119902minus1)) = 119896

where 119896 is constantEquation (2) of order 119902 with two independent variables

which admits a symmetry 119883 that is associated with aconserved vector 119879 is reduced to an ODE of order 119902 minus 1namely 119879119903 = 119896 where 119879

119903 is given by (26) for solutionsinvariant under119883

4 Main Results

Firstly we use the nonlocal conservation method given byIbragimov Equation (1) admits the following three Lie-pointsymmetry generators [12]

1198831=

120597

120597119909

1198832=

120597

120597119905

1198833= 119905

120597

120597119905

minus 119906

120597

120597119906

(27)

Equation (1) does not have the usual Lagrangian TheLagrangian for (1) is

119871 = 119908 (119906119905minus 2119906119909119906119909119909minus 119906119909119909119909

119906 minus 119906119909119909119905) = 0 (28)

The adjoint equation for (1) is

119864lowast(119905 119909 119906 119908 119908

119909119909119909119909)

=

120575

120575119906

[119908 (119906119905minus 2119906119909119906119909119909minus 119906119909119909119909

119906 minus 119906119909119909119905)]

(29)

and we can get the adjoint equation

119864lowast= 119908119909119906119909119909minus 119908119905+ 119908119909119909119906119909+ 119908119909119909119909

119906 + 119908119909119909119905

= 0 (30)

where 119908 is the adjoint variable Let us investigate the quasi-self-adjointness of (1) We make the ansatz of 119908 = 120601(119906)Taking into account (29) of 119864lowast and using (16) together withits consequences 119908 = 120601(119906) 119908

119905= 1206011015840119906119905 119908119909= 1206011015840119906119909 119908119909119909

=

120601101584010158401199062

119909+1206011015840119906119909119909119908119909119909119909

= 1206011015840101584010158401199063

119909+1206011015840119906119909119909119909

+312060110158401015840119906119909119906119909119909 and 119908

119909119909119905=

1206011015840101584010158401199061199051199062

119909+212060110158401015840119906119909119906119909119905+12060110158401015840119906119905119906119909119909+1206011015840119906119909119909119905

we rewrite (30) in thefollowing form

212060110158401015840119906119909119906119909119909minus 1206011015840119906119905+ 120601101584010158401199063

119909+ 1199061206011015840101584010158401199063

119909+ 3119906120601

10158401015840119906119909119906119909119909

+ 1199061206011015840119906119909119909119909

+ 1206011015840101584010158401199061199051199062

119909+ 212060110158401015840119906119909119906119909119905

+ 12060110158401015840119906119905119906119909119909+ 1206011015840119906119909119909119905

= 120582 (119906119905minus 2119906119909119906119909119909minus 119906119909119909119909

119906 minus 119906119909119909119905)

(31)

Equation (31) should be satisfied identically in all vari-ables 119906

119905 119906119909 119906119909119909 Comparing the coefficients of 119906

119905in both

sides of (31) we can easily obtain 120582 = minus1206011015840 Then we equate all

coefficients of linear and nonlinear mixed derivatives termsand get 120601(119906) = 119888

1119906 + 1198882

4 Advances in Mathematical Physics

The conserved components of (1) associated with asymmetry can be obtained from (14) as follows

119879119905= 120585119905119871 +119882(

120597119871

120597119906119905

+ 1198632

119909

120597119871

120597119906119909119909119905

) + 119863119909(119882)(minus119863

119909

120597119871

120597119906119909119909119905

)

+ 1198632

119909(119882)(

120597119871

120597119906119909119909119905

)

119879119909= 120585119909119871 +119882(

120597119871

120597119906119909

minus 119863119909

120597119871

120597119906119909119909

+ 1198632

119909

120597119871

120597119906119909119909119909

+ 119863119909119863119905

120597119871

120597119906119905119909119909

)

+ 119863119909(119882)(

120597119871

120597119906119909119909

minus 119863119909

120597119871

120597119906119909119909119909

minus 119863119905

120597119871

120597119906119905119909119909

)

+ 119863119905(119882)(minus119863

119909

120597119871

120597119906119905119909119909

) + 119863119909119863119905(119882)(

120597119871

120597119906119909119909119905

)

+ 1198632

119909(119882)(

120597119871

120597119906119909119909119909

)

(32)

where119882 is Lie characteristic function According to (31) wecan determine 119908 at two cases 119888

1= 1 1198882= 0 and 119888

1= 0 1198882= 1

has an infinite number of solutions The conservation lawsassociated with the generators (27) are below Firstly we take119908 = 119906

Case 1 Now let us make calculations for the operator 1198831=

120597120597119909 in detail For this operator the infinitesimals are 120585119909 =1 120585119905 = 0 and 120578 = 0 and we get 119882 = minus119906

119909and the

corresponding conserved vector of (1) as

119879119909

1= minus119906119905119906

119879119905

1= 119906119909119906

(33)

It is readily seen that in this case we obtain null conservedvectors by the definition of conservation laws

Case 2 In this case for the generator1198832= 120597120597119905 (120585119909 = 0 120585119905 =

1 and 120578 = 0) we calculate 119882 = minus119906119905and the conserved

quantities of (1) as

119879119909

2= 2119906119905119906119909119909119906 minus 119906119905119905119906119909+ 1199061199051199091199091199062+ 119906119905119905119909119906

119879119905

2= minus 2119906

119909119906119909119909119906 minus 1199062119906119909119909119909

+ 119906119905119906119909119909minus 119906119905119909119906119909

(34)

The divergence condition becomes

119863119905119879119905+ 119863119909119879119909= minus1199062

119905119909+ 119906119905119905119909119909

119906 minus 119906119905119905119909119906119909+ 119906119905119909119909119906119905 (35)

We observe that extra terms emerge By some adjustmentsthese terms can be absorbed as

119863119905119879119905+ 119863119909119879119909= 119863119905(minus119906119905119909119906119909+ 119906119905119909119909119906) (36)

into the conservation law Taking these terms across andincluding them into the conserved flows we get

119879119909

2= 2119906119905119906119909119909119906 minus 119906119905119905119906119909+ 1199061199051199091199091199062+ 119906119905119905119909119906

119879119905

2= minus2119906

119909119906119909119909119906 minus 1199062119906119909119909119909

+ 119906119905119906119909119909minus 119906119905119909119909119906

(37)

The modified conserved quantities are now labeled 119879119894 where

119863119905(119879119905) + 119863

119909(119879119909) = 0 modulo the equation It is readily

seen that in this case we obtain null conserved vectors by thedefinition of conservation laws

Case 3 Let us find the conservation law provided by 1198833=

119905(120597120597119905) minus 119906(120597120597119906) (120585119909 = 0 120585119905 = 119905 and 120578 = minus119906) In this case wehave 119882 = minus119906 minus 119905119906

119905and (32) yield the conservation laws (9)

with

119879119909

3= 31199062119906119909119909+ 3119906119906

119905119909minus 3119906119905119906119909+ 2119905119906119906

119905119906119909119909

minus 119905119906119909119906119905119905+ 1199051199062119906119905119909119909

+ 119905119906119906119905119905119909

119879119905

3= minus2119905119906119906

119909119906119909119909minus 1199051199062119906119909119909119909

minus 1199062+ 2119906119906

119909119909

+ 119905119906119905119906119909119909+ 1199062

119909+ 119905119906119909119906119905119909

(38)

The divergence of (38) is

119863119905119879119905+ 119863119909119879119909= 119863119909(119905119906119905119906119905119909+ 3119906119906

119905119909+ 119905119906119906119909119909119905) (39)

After some adjustments the nontrivial conserved quantitiesare as follows

119879119909

3= 31199062119906119909119909minus 3119906119905119906119909+ 2119905119906119906

119905119906119909119909

minus 119905119906119909119906119905119905+ 1199051199062119906119905119909119909

minus 119905119906119905119906119905119909

119879119905

3= minus2119905119906119906

119909119906119909119909minus 1199051199062119906119909119909119909

minus 1199062+ 2119906119906

119909119909

+ 119905119906119905119906119909119909+ 1199062

119909+ 119905119906119909119906119905119909

(40)

For the second case 119908(119909 119905) = 1 the corresponding con-servation laws are as follows

Case 4 For the generator 1198831= 120597120597119909 and Lie characteristic

function119882 = minus119906119909 we get the following conserved vectors

119879119909

4= 119906119905

119879119905

4= minus119906119909

(41)

Again like in Case 1 we obtain the null conserved vectors

Case 5 In this case for the generator 1198832= 120597120597119905 (120585119909 = 0

120585119905= 1 and 120578 = 0) we calculate119882 = minus119906

119905and the conserved

quantities of (1) as

119879119909

5= 119906119905119906119909119909+ 119906119905119909119906119909+ 119906119905119909119909119906 + 119906119905119905119909

119879119905

5= minus 2119906

119909119906119909119909minus 119906119906119909119909119909

(42)

After adjustment according to divergence we get modifiedconserved vectors

119879119909

5= 119906119905119906119909119909+ 119906119905119909119906119909+ 119906119905119909119909119906 + 119906119905119905119909

119879119905

5= minus 2119906

119909119906119909119909minus 119906119906119909119909119909

minus 119906119909119909119905

(43)

Again like in Case 2 we obtain the null conserved vectors

Advances in Mathematical Physics 5

Case 6 Lastly we consider the generator 1198833= 119905(120597120597119905) minus

119906(120597120597119906) where 120585119909 = 0 120585119905 = 119905 and 120578 = minus119906 In this case wehave 119882 = minus119906 minus 119905119906

119905and (32) yield the conservation laws (9)

with

119879119909

6= 2119906119906

119909119909+ 119905119906119909119909119906119905+ 1199062

119909+ 119905119906119909119906119905119909

+ 119906119905119906119905119909119909

+ 2119906119905119909+ 119905119906119905119905119909

119879119905

6= minus2119905119906

119909119906119909119909minus 119906119905119906119909119909119909

minus 119906 + 119906119909119909

(44)

We calculate the divergence

119863119905119879119905+ 119863119909119879119909= 119863119905(119905119906119909119909119905

+ 119906119909119909) (45)

Following the same line we find that the modified nontrivialconserved vectors are

119879119909

6= 2119906119906

119909119909+ 119905119906119909119909119906119905+ 1199062

119909+ 119905119906119909119906119905119909

+ 119905119906119906119905119909119909

+ 2119906119905119909+ 119905119906119905119905119909

119879119905

6= minus2119905119906

119909119906119909119909minus 119905119906119906119909119909119909

minus 119906 minus 119905119906119909119909119905

(46)

Now we will derive the conservation laws of the MHSequation by themultipliermethodThe third ordermultiplierfor (1) is Λ(119909 119905 119906 119906

119909 119906119909119909 119906119909119909119909

119906119909119909119905) and the corresponding

determining equation is

120575

120575119906

[Λ (119906119905minus 2119906119909119906119909119909minus 119906119909119909119909

119906 minus 119906119909119909119905)] = 0 (47)

Expanding and then separating (47) with respect to differentcombinations of derivatives of 119906 yields the following overde-termined system for the multipliers

Λ119906119906= 0 Λ

119905= 0 Λ

119909= 0 Λ

119906119909

= 0

Λ119906119909119909

= 0 Λ119906119909119909119909

= 0 Λ119906119905119909119909

= 0

(48)

The solution of system (48) can be expressed as

Λ = 1198881119906 + 1198882 (49)

where 1198881 1198882are constants Corresponding to the above

multiplier we have the following conserved vectors of (49)

119879119909

1= minus 119906

2119906119909119909minus

119906119905119909119906

2

+

119906119909119906119905

2

119879119905

1=

1

2

1199062minus

1

2

119906119909119909119906

119879119909

2= minus 119906

119909119909119906 minus

1199062

119909

2

119879119905

2= 119906 minus 119906

119909119909

(50)

The multiplier approach gave two local conservation laws forthe MHS equation

Now we will derive the exact group-invariant solutionof (1) using the relationship between local conservation lawsand Lie-point symmetries Equation (1) admits the symmetrygenerators 119883

1= 120597120597119909 119883

2= 120597120597119905 associated with the

conservation law

119863119905(119906 minus 119906

119909119909) + 119863119909(minus119906119909119909119906 minus

1199062

119909

2

) = 0 (51)

We set 119883 = 1198831+ 1205721198832 Then the canonical coordinates of 119883

are 119904 = 119909 119903 = 120572119909 minus 119905 and 119906 Since 119879 = (119879119903 119879119904) is associated

with119883 we have to find the value of 119879119903 Using (26) we obtainthe following conserved vector

119879119903= 119896 = 120572

2119906119906119903119903+

1205722

2

1199062

119903 (52)

We can substitute the variables 119906119903= 119901 and 119906

119903119903= 119901(119889119901119889119906)

in (52) After using these variables (52) reduces to first orderordinary differential equation (ODE)

119901 (119906) = plusmn

radic119906 (2119896119906 + 11988811205722)

119906120572

(53)

We can solve (53) by separation of variables and the solutiongives rise to

119903 + 1198882

= plusmn

120572

2119896

radic21198961199062+ 12057221199061198881∓

1

8

times ((12057231198881ln(1

2

((12) 12057221198881+ 2119896119906)radic2

radic119896

+radic21198961199062+ 12057221199061198881)radic2) times (119896

32)

minus1

)

119903 = 120572119909 minus 119905

(54)

which constitutes the solution of the MHS equation

5 Conclusion

In this work we studied conservation laws symmetry reduc-tions and exact solutions of MHS equation Utilizing nonlo-cal conservation and multiplier method we constructed fourdistinct local conservation laws (see (40) (46) and (50)) It isclear that by using Ibragimovrsquos nonlocal conservationmethodone can obtain infinite nonlocal conservation laws Thenusing the double reduction method we reduced the MHSequation to second order ODE in the canonical variables (see(52)) Exact group-invariant solutions were constructed byintegrating the reduced ODE

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by Eskisehir Osmangazi UniversityScientific Research Projects (Grant no 2013-281)

6 Advances in Mathematical Physics

References

[1] E Noether ldquoInvariante Variationsproblemerdquo Nachrichten vonder koniglichen Gesellschaft der Wissenschaften zu Gottingenvol 2 pp 235ndash257 1918 English translation in TransportTheoryand Statistical Physics vol 1 no 3 pp 186ndash207 1971

[2] H Steudel ldquoUber die Zuordnung zwischen Invarianzeigen-schaften und Erhaltungssatzenrdquo Zeitschrift fur Naturforschungvol 17 pp 129ndash132 1962

[3] P J Olver Application of Lie groups to Differential EquationsSpringer New York NY USA 1993

[4] A H Kara and FMMahomed ldquoRelationship between symme-tries and conservation lawsrdquo International Journal ofTheoreticalPhysics vol 39 no 1 pp 23ndash40 2000

[5] S C Anco and G Bluman ldquoDirect construction method forconservation laws of partial differential equations I Examplesof conservation law classificationsrdquo European Journal of AppliedMathematics vol 13 no 5 pp 545ndash566 2002

[6] A H Kara and FMMahomed ldquoNoether-type symmetries andconservation laws via partial Lagrangiansrdquo Nonlinear Dynam-ics vol 45 no 3-4 pp 367ndash383 2006

[7] N H Ibragimov ldquoA new conservation theoremrdquo Journal ofMathematical Analysis and Applications vol 333 no 1 pp 311ndash328 2007

[8] R Naz F M Mahomed and D P Mason ldquoComparison ofdifferent approaches to conservation laws for some partialdifferential equations in fluid mechanicsrdquo Applied Mathematicsand Computation vol 205 no 1 pp 212ndash230 2008

[9] A F Cheviakov ldquoComputation of fluxes of conservation lawsrdquoJournal of EngineeringMathematics vol 66 no 1ndash3 pp 153ndash1732010

[10] T Wolf ldquoA comparison of four approaches to the calculation ofconservation lawsrdquo European Journal of Applied Mathematicsvol 13 no 2 pp 129ndash152 2002

[11] J K Hunter and R Saxton ldquoDynamics of director fieldsrdquo SIAMJournal on Applied Mathematics vol 51 no 6 pp 1498ndash15211991

[12] A G Johnpillai and C M Khalique ldquoSymmetry reductionsexact solutions and conservation laws of a modified Hunter-Saxton equationrdquo Abstract and Applied Analysis vol 2013Article ID 204746 5 pages 2013

[13] B Khesin and G Misiołek ldquoEuler equations on homogeneousspaces and Virasoro orbitsrdquo Advances in Mathematics vol 176no 1 pp 116ndash144 2003

[14] J K Hunter and Y X Zheng ldquoOn a completely integrable non-linear hyperbolic variational equationrdquo Physica D NonlinearPhenomena vol 79 no 2ndash4 pp 361ndash386 1994

[15] M Nadjafikhah and F Ahangari ldquoSymmetry analysis andconservation laws for the Hunter-Saxton equationrdquo Communi-cations in Theoretical Physics vol 59 no 3 pp 335ndash348 2013

[16] A Sjoberg ldquoDouble reduction of PDEs from the association ofsymmetries with conservation laws with applicationsrdquo AppliedMathematics and Computation vol 184 no 2 pp 608ndash6162007

[17] K R Adem andCM Khalique ldquoExact solutions and conserva-tion laws of a (2+1)-dimensional nonlinear KP-BBM equationrdquoAbstract and Applied Analysis vol 2013 Article ID 791863 5pages 2013

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Advances in Mathematical Physics

2 Preliminaries

We briefly present notation to be used and recall basic defini-tions and theorems which utilize below [2 7 16] Considerthe 119896th-order system of PDEs of 119899 independent variables119909 = (119909

1 1199092 119909

119899) and 119898 dependent variables 119906 = (119906

1

1199062 119906

119898)

119864120572(119909 119906 119906

(1) 119906

(119896)) = 0 120572 = 1 119898 (2)

where 119906(119894)

is the collection of 119894th-order partial derivatives119906120572

119894= 119863119894(119906120572) 119906120572119894119895= 119863119895119863119894(119906120572) respectively with the total

differentiation operator with respect to 119909119894 given by

119863119894=

120597

120597119909119894+ 119906120572

119894

120597

120597119906120572+ 119906120572

119894119895

120597

120597119906120572

119895

+ sdot sdot sdot 119894 = 1 119899 (3)

in which the summation convention is used The Lie-pointgenerator is

119883 = 120585119894 120597

120597119909119894

+ 120578120572 120597

120597119906120572 (4)

where 120585119894 and 120578

120572 are functions of only independent anddependent functionsThe operator (4) is an abbreviated formof the infinite formal sum

119883 = 120585119894 120597

120597119909119894

+ 120578120572 120597

120597119906120572+sum

119904ge1

120577120572

11989411198942sdotsdotsdot119894119904

120597

120597119906120572

11989411198942sdotsdotsdot119894119904

(5)

where the additional coefficients can be determined from theprolongation formulae

120577120572

119894= 119863119894(120578120572) minus 120585119895119906120572

119895119894

120577120572

1198941sdotsdotsdot119894119904

= 1198631198941

sdot sdot sdot 119863119894119904

(120577120572

1198941sdotsdotsdot119894119904minus1

) minus 120585119895119906120572

1198951198941sdotsdotsdot119894119904

119904 gt 1

(6)

The Noether operators associated with a Lie-point generator119883 are

119873119894= 120585119894+119882120572 120575

120575119906120572

119894

+

infin

sum

119904ge1

1198631198941

sdot sdot sdot 119863119894119904

120597

120597119906120572

1198941sdotsdotsdot119894119904

119894 = 1 2 119899

(7)

in which119882120572 is the Lie characteristic function

119882120572= 120578120572minus 120585119895119906120572

119895 (8)

The conserved vector of (2) where each119879119894 isin 119860119860 is the spaceof all differential functions satisfies the equation

119863119894119879119894

|(2)= 0 (9)

along the solution of (2)

3 Conservation Laws Methods

31 Nonlocal Conservation Method We will denote indepen-dent variables119909 = (119909

1 1199092)with1199091 = 1199091199092 = 119905 one dependent

variable 119906 together with its derivatives up to119901 arbitrary orderThe 119901th-order PDE

119864 (119909 119906 1199061 119906

119901) = 0 (10)

has always formal Lagrangian Formal Lagrangian is mul-tiplication of a new adjoint variable 119908(119909 119905) with a givenequation Namely

119871 = 119908120572119864120572 (11)

With this formal Lagrangian

119864lowast=

120575119871

120575119906120572

(12)

adjoint equation is constructed Here 120575120575119906 is the Euler-Lagrange operator and defined by

120575

120575119906120572=

120597

120597119906120572+

infin

sum

119904ge1

(minus1)1199041198631198941

sdot sdot sdot 119863119894119904

120597

120597119906120572

1198941sdotsdotsdot119894119904

120572 = 1 119898

(13)

Theorem 1 (see [7]) Every Lie-point Lie-Backlund and non-local symmetry of (2) gives a conservation law for the equationunder consideration The conserved vector components aredetermined with

119879119894= 120585119894119871 +119882

120572[

120597119871

120597119906119894

minus 119863119895(

120597119871

120597119906119894119895

) + 119863119895119863119896(

120597119871

120597119906119894119895119896

)

minus119863119895119863119896119863119898(

120597119871

120597119906119894119895119896119898

)]

+ 119863119895(119882120572) [

120597119871

120597119906119894119895

minus 119863119896(

120597119871

120597119906119894119895119896

) + 119863119896119863119898(

120597119871

120597119906119894119895119896119898

)]

+ 119863119895119863119896(119882120572) [

120597119871

120597119906119894119895119896

minus 119863119898(

120597119871

120597119906119894119895119896119898

)]

+ 119863119895119863119896119863119898(119882120572) [(

120597119871

120597119906119894119895119896119898

)]

(14)

where Lagrangian (formal Lagrangian) function is given by

119871 = 119908120572119864120572(119909 119906 119906

(1) 119906

(119896)) (15)

120585119894 120578120572 are the coefficient functions of the associated generator(4)

The conserved vectors obtained from (14) involve thearbitrary solutions 119908 of the adjoint equation (12) and henceone obtains an infinite number of conservation laws for (1) bychoosing 119908

Definition 2 We say that (2) is strictly self-adjoint if theadjoint equation (12) becomes equivalent to (2) after the sub-stitution 119908 = 119906

120575119871

120575119906120572= 120582119864 (119909 119905 119906 119906

119909 119906

119909119909119905) (16)

with 120582 being generic coefficient

Advances in Mathematical Physics 3

Definition 3 We say that (2) is quasi-self-adjoint if the adjointequation (12) becomes equivalent to (2) after the substitution119908 = 120601(119906) 120601(119906) = 0

32 The Multiplier Method Amultiplier Λ120572(119909 119906 119906

119909 ) has

the property that

Λ120572119864120572= 119863119894119879119894 (17)

holds identically Here we will consider multipliers of thirdorder that is Λ

120572= Λ

120572(119909 119905 119906 119906

119909 119906119909119909 119906119909119909119909

) The righthand side of (17) is a divergence expressionThe determiningequation for the multiplier Λ

120572is

120575 (Λ120572119864120572)

120575119906120572

= 0 (18)

Once the multipliers are obtained the conserved vectors arecalculated via a homotopy formula [5 17] All the multiplierscan be calculated with the aid of (18) for which the equationcan be expressed as a local conservation law [9]

33 Double Reduction Method Let 119883 be any Lie-point sym-metry and 119879119894 are the components of conserved vector If 119883and 119879 satisfy

119883(119879119894) + 119879119894119863119895(120585119895) minus 119879119895119863119895(120585119894) = 0 119894 = 1 2 (19)

then 119883 is associated with 119879 We define a nonlocal variable Vby 119879119905 = V

119909 119879119909 = minusV

119905 Taking the similarity variables 119903 119904 120603

with the generator119883 = 120597120597119904 we have in similarity variables

119879119903= V119904 119879

119904= minusV119903 (20)

so that the conservation law is rewritten as

119863119903119879119903+ 119863119904119879119904= 0 (21)

Using the chain rule we have

119863119909= 119863119909(119904)119863119904+ 119863119909(119903)119863119903 119863

119905= 119863119905(119904)119863119904+ 119863119905(119903)119863119903

(22)

so that

V119909= V119904119863119909(119904) + V

119903119863119909(119903) V

119905= V119904119863119905(119904) + V

119903119863119905(119903) (23)

and so

119879119905= 119879119903119863119909(119904) minus 119879

119904119863119909(119903) 119879

119909= 119879119903119863119909(119904) minus 119879

119904119863119905(119903)

(24)

Using the above linear algebraical system we can get

119879119904=

119879119905119863119905(119904) + 119879

119909119863119909(119904)

119863119905(119903)119863119909(119904) minus 119863

119909(119903)119863119905(119904)

(25)

119879119903=

119879119905119863119905(119903) + 119879

119909119863119909(119903)

119863119905(119903)119863119909(119904) minus 119863

119909(119903)119863119905(119904)

(26)

The components 119879119909 119879119905 depend on (119905 119909 119906 119906(1) 119906(2)

119906(119902minus1)

) which means that 119879119904 119879119903 depend on (119904 119903 120603 120603

119903

120603119903119903 120603

119903(119902minus1)) for solutions invariant under 119883 Therefore

(21) becomes (120597119879119904120597119904) + 119863119903119879119903= 0

For 119879 associated with 119883 we have 119883119879119903 = 0 and 119883119879119904 = 0Thus 119879119903 and 119879119903 are invariant under 119883 This means 120597119879119904120597119904 =0 and 120597119879

119903120597119904 = 0 so that 119879119903(119903 120603 120603

119903 120603119903119903 120603

119903(119902minus1)) = 119896

where 119896 is constantEquation (2) of order 119902 with two independent variables

which admits a symmetry 119883 that is associated with aconserved vector 119879 is reduced to an ODE of order 119902 minus 1namely 119879119903 = 119896 where 119879

119903 is given by (26) for solutionsinvariant under119883

4 Main Results

Firstly we use the nonlocal conservation method given byIbragimov Equation (1) admits the following three Lie-pointsymmetry generators [12]

1198831=

120597

120597119909

1198832=

120597

120597119905

1198833= 119905

120597

120597119905

minus 119906

120597

120597119906

(27)

Equation (1) does not have the usual Lagrangian TheLagrangian for (1) is

119871 = 119908 (119906119905minus 2119906119909119906119909119909minus 119906119909119909119909

119906 minus 119906119909119909119905) = 0 (28)

The adjoint equation for (1) is

119864lowast(119905 119909 119906 119908 119908

119909119909119909119909)

=

120575

120575119906

[119908 (119906119905minus 2119906119909119906119909119909minus 119906119909119909119909

119906 minus 119906119909119909119905)]

(29)

and we can get the adjoint equation

119864lowast= 119908119909119906119909119909minus 119908119905+ 119908119909119909119906119909+ 119908119909119909119909

119906 + 119908119909119909119905

= 0 (30)

where 119908 is the adjoint variable Let us investigate the quasi-self-adjointness of (1) We make the ansatz of 119908 = 120601(119906)Taking into account (29) of 119864lowast and using (16) together withits consequences 119908 = 120601(119906) 119908

119905= 1206011015840119906119905 119908119909= 1206011015840119906119909 119908119909119909

=

120601101584010158401199062

119909+1206011015840119906119909119909119908119909119909119909

= 1206011015840101584010158401199063

119909+1206011015840119906119909119909119909

+312060110158401015840119906119909119906119909119909 and 119908

119909119909119905=

1206011015840101584010158401199061199051199062

119909+212060110158401015840119906119909119906119909119905+12060110158401015840119906119905119906119909119909+1206011015840119906119909119909119905

we rewrite (30) in thefollowing form

212060110158401015840119906119909119906119909119909minus 1206011015840119906119905+ 120601101584010158401199063

119909+ 1199061206011015840101584010158401199063

119909+ 3119906120601

10158401015840119906119909119906119909119909

+ 1199061206011015840119906119909119909119909

+ 1206011015840101584010158401199061199051199062

119909+ 212060110158401015840119906119909119906119909119905

+ 12060110158401015840119906119905119906119909119909+ 1206011015840119906119909119909119905

= 120582 (119906119905minus 2119906119909119906119909119909minus 119906119909119909119909

119906 minus 119906119909119909119905)

(31)

Equation (31) should be satisfied identically in all vari-ables 119906

119905 119906119909 119906119909119909 Comparing the coefficients of 119906

119905in both

sides of (31) we can easily obtain 120582 = minus1206011015840 Then we equate all

coefficients of linear and nonlinear mixed derivatives termsand get 120601(119906) = 119888

1119906 + 1198882

4 Advances in Mathematical Physics

The conserved components of (1) associated with asymmetry can be obtained from (14) as follows

119879119905= 120585119905119871 +119882(

120597119871

120597119906119905

+ 1198632

119909

120597119871

120597119906119909119909119905

) + 119863119909(119882)(minus119863

119909

120597119871

120597119906119909119909119905

)

+ 1198632

119909(119882)(

120597119871

120597119906119909119909119905

)

119879119909= 120585119909119871 +119882(

120597119871

120597119906119909

minus 119863119909

120597119871

120597119906119909119909

+ 1198632

119909

120597119871

120597119906119909119909119909

+ 119863119909119863119905

120597119871

120597119906119905119909119909

)

+ 119863119909(119882)(

120597119871

120597119906119909119909

minus 119863119909

120597119871

120597119906119909119909119909

minus 119863119905

120597119871

120597119906119905119909119909

)

+ 119863119905(119882)(minus119863

119909

120597119871

120597119906119905119909119909

) + 119863119909119863119905(119882)(

120597119871

120597119906119909119909119905

)

+ 1198632

119909(119882)(

120597119871

120597119906119909119909119909

)

(32)

where119882 is Lie characteristic function According to (31) wecan determine 119908 at two cases 119888

1= 1 1198882= 0 and 119888

1= 0 1198882= 1

has an infinite number of solutions The conservation lawsassociated with the generators (27) are below Firstly we take119908 = 119906

Case 1 Now let us make calculations for the operator 1198831=

120597120597119909 in detail For this operator the infinitesimals are 120585119909 =1 120585119905 = 0 and 120578 = 0 and we get 119882 = minus119906

119909and the

corresponding conserved vector of (1) as

119879119909

1= minus119906119905119906

119879119905

1= 119906119909119906

(33)

It is readily seen that in this case we obtain null conservedvectors by the definition of conservation laws

Case 2 In this case for the generator1198832= 120597120597119905 (120585119909 = 0 120585119905 =

1 and 120578 = 0) we calculate 119882 = minus119906119905and the conserved

quantities of (1) as

119879119909

2= 2119906119905119906119909119909119906 minus 119906119905119905119906119909+ 1199061199051199091199091199062+ 119906119905119905119909119906

119879119905

2= minus 2119906

119909119906119909119909119906 minus 1199062119906119909119909119909

+ 119906119905119906119909119909minus 119906119905119909119906119909

(34)

The divergence condition becomes

119863119905119879119905+ 119863119909119879119909= minus1199062

119905119909+ 119906119905119905119909119909

119906 minus 119906119905119905119909119906119909+ 119906119905119909119909119906119905 (35)

We observe that extra terms emerge By some adjustmentsthese terms can be absorbed as

119863119905119879119905+ 119863119909119879119909= 119863119905(minus119906119905119909119906119909+ 119906119905119909119909119906) (36)

into the conservation law Taking these terms across andincluding them into the conserved flows we get

119879119909

2= 2119906119905119906119909119909119906 minus 119906119905119905119906119909+ 1199061199051199091199091199062+ 119906119905119905119909119906

119879119905

2= minus2119906

119909119906119909119909119906 minus 1199062119906119909119909119909

+ 119906119905119906119909119909minus 119906119905119909119909119906

(37)

The modified conserved quantities are now labeled 119879119894 where

119863119905(119879119905) + 119863

119909(119879119909) = 0 modulo the equation It is readily

seen that in this case we obtain null conserved vectors by thedefinition of conservation laws

Case 3 Let us find the conservation law provided by 1198833=

119905(120597120597119905) minus 119906(120597120597119906) (120585119909 = 0 120585119905 = 119905 and 120578 = minus119906) In this case wehave 119882 = minus119906 minus 119905119906

119905and (32) yield the conservation laws (9)

with

119879119909

3= 31199062119906119909119909+ 3119906119906

119905119909minus 3119906119905119906119909+ 2119905119906119906

119905119906119909119909

minus 119905119906119909119906119905119905+ 1199051199062119906119905119909119909

+ 119905119906119906119905119905119909

119879119905

3= minus2119905119906119906

119909119906119909119909minus 1199051199062119906119909119909119909

minus 1199062+ 2119906119906

119909119909

+ 119905119906119905119906119909119909+ 1199062

119909+ 119905119906119909119906119905119909

(38)

The divergence of (38) is

119863119905119879119905+ 119863119909119879119909= 119863119909(119905119906119905119906119905119909+ 3119906119906

119905119909+ 119905119906119906119909119909119905) (39)

After some adjustments the nontrivial conserved quantitiesare as follows

119879119909

3= 31199062119906119909119909minus 3119906119905119906119909+ 2119905119906119906

119905119906119909119909

minus 119905119906119909119906119905119905+ 1199051199062119906119905119909119909

minus 119905119906119905119906119905119909

119879119905

3= minus2119905119906119906

119909119906119909119909minus 1199051199062119906119909119909119909

minus 1199062+ 2119906119906

119909119909

+ 119905119906119905119906119909119909+ 1199062

119909+ 119905119906119909119906119905119909

(40)

For the second case 119908(119909 119905) = 1 the corresponding con-servation laws are as follows

Case 4 For the generator 1198831= 120597120597119909 and Lie characteristic

function119882 = minus119906119909 we get the following conserved vectors

119879119909

4= 119906119905

119879119905

4= minus119906119909

(41)

Again like in Case 1 we obtain the null conserved vectors

Case 5 In this case for the generator 1198832= 120597120597119905 (120585119909 = 0

120585119905= 1 and 120578 = 0) we calculate119882 = minus119906

119905and the conserved

quantities of (1) as

119879119909

5= 119906119905119906119909119909+ 119906119905119909119906119909+ 119906119905119909119909119906 + 119906119905119905119909

119879119905

5= minus 2119906

119909119906119909119909minus 119906119906119909119909119909

(42)

After adjustment according to divergence we get modifiedconserved vectors

119879119909

5= 119906119905119906119909119909+ 119906119905119909119906119909+ 119906119905119909119909119906 + 119906119905119905119909

119879119905

5= minus 2119906

119909119906119909119909minus 119906119906119909119909119909

minus 119906119909119909119905

(43)

Again like in Case 2 we obtain the null conserved vectors

Advances in Mathematical Physics 5

Case 6 Lastly we consider the generator 1198833= 119905(120597120597119905) minus

119906(120597120597119906) where 120585119909 = 0 120585119905 = 119905 and 120578 = minus119906 In this case wehave 119882 = minus119906 minus 119905119906

119905and (32) yield the conservation laws (9)

with

119879119909

6= 2119906119906

119909119909+ 119905119906119909119909119906119905+ 1199062

119909+ 119905119906119909119906119905119909

+ 119906119905119906119905119909119909

+ 2119906119905119909+ 119905119906119905119905119909

119879119905

6= minus2119905119906

119909119906119909119909minus 119906119905119906119909119909119909

minus 119906 + 119906119909119909

(44)

We calculate the divergence

119863119905119879119905+ 119863119909119879119909= 119863119905(119905119906119909119909119905

+ 119906119909119909) (45)

Following the same line we find that the modified nontrivialconserved vectors are

119879119909

6= 2119906119906

119909119909+ 119905119906119909119909119906119905+ 1199062

119909+ 119905119906119909119906119905119909

+ 119905119906119906119905119909119909

+ 2119906119905119909+ 119905119906119905119905119909

119879119905

6= minus2119905119906

119909119906119909119909minus 119905119906119906119909119909119909

minus 119906 minus 119905119906119909119909119905

(46)

Now we will derive the conservation laws of the MHSequation by themultipliermethodThe third ordermultiplierfor (1) is Λ(119909 119905 119906 119906

119909 119906119909119909 119906119909119909119909

119906119909119909119905) and the corresponding

determining equation is

120575

120575119906

[Λ (119906119905minus 2119906119909119906119909119909minus 119906119909119909119909

119906 minus 119906119909119909119905)] = 0 (47)

Expanding and then separating (47) with respect to differentcombinations of derivatives of 119906 yields the following overde-termined system for the multipliers

Λ119906119906= 0 Λ

119905= 0 Λ

119909= 0 Λ

119906119909

= 0

Λ119906119909119909

= 0 Λ119906119909119909119909

= 0 Λ119906119905119909119909

= 0

(48)

The solution of system (48) can be expressed as

Λ = 1198881119906 + 1198882 (49)

where 1198881 1198882are constants Corresponding to the above

multiplier we have the following conserved vectors of (49)

119879119909

1= minus 119906

2119906119909119909minus

119906119905119909119906

2

+

119906119909119906119905

2

119879119905

1=

1

2

1199062minus

1

2

119906119909119909119906

119879119909

2= minus 119906

119909119909119906 minus

1199062

119909

2

119879119905

2= 119906 minus 119906

119909119909

(50)

The multiplier approach gave two local conservation laws forthe MHS equation

Now we will derive the exact group-invariant solutionof (1) using the relationship between local conservation lawsand Lie-point symmetries Equation (1) admits the symmetrygenerators 119883

1= 120597120597119909 119883

2= 120597120597119905 associated with the

conservation law

119863119905(119906 minus 119906

119909119909) + 119863119909(minus119906119909119909119906 minus

1199062

119909

2

) = 0 (51)

We set 119883 = 1198831+ 1205721198832 Then the canonical coordinates of 119883

are 119904 = 119909 119903 = 120572119909 minus 119905 and 119906 Since 119879 = (119879119903 119879119904) is associated

with119883 we have to find the value of 119879119903 Using (26) we obtainthe following conserved vector

119879119903= 119896 = 120572

2119906119906119903119903+

1205722

2

1199062

119903 (52)

We can substitute the variables 119906119903= 119901 and 119906

119903119903= 119901(119889119901119889119906)

in (52) After using these variables (52) reduces to first orderordinary differential equation (ODE)

119901 (119906) = plusmn

radic119906 (2119896119906 + 11988811205722)

119906120572

(53)

We can solve (53) by separation of variables and the solutiongives rise to

119903 + 1198882

= plusmn

120572

2119896

radic21198961199062+ 12057221199061198881∓

1

8

times ((12057231198881ln(1

2

((12) 12057221198881+ 2119896119906)radic2

radic119896

+radic21198961199062+ 12057221199061198881)radic2) times (119896

32)

minus1

)

119903 = 120572119909 minus 119905

(54)

which constitutes the solution of the MHS equation

5 Conclusion

In this work we studied conservation laws symmetry reduc-tions and exact solutions of MHS equation Utilizing nonlo-cal conservation and multiplier method we constructed fourdistinct local conservation laws (see (40) (46) and (50)) It isclear that by using Ibragimovrsquos nonlocal conservationmethodone can obtain infinite nonlocal conservation laws Thenusing the double reduction method we reduced the MHSequation to second order ODE in the canonical variables (see(52)) Exact group-invariant solutions were constructed byintegrating the reduced ODE

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by Eskisehir Osmangazi UniversityScientific Research Projects (Grant no 2013-281)

6 Advances in Mathematical Physics

References

[1] E Noether ldquoInvariante Variationsproblemerdquo Nachrichten vonder koniglichen Gesellschaft der Wissenschaften zu Gottingenvol 2 pp 235ndash257 1918 English translation in TransportTheoryand Statistical Physics vol 1 no 3 pp 186ndash207 1971

[2] H Steudel ldquoUber die Zuordnung zwischen Invarianzeigen-schaften und Erhaltungssatzenrdquo Zeitschrift fur Naturforschungvol 17 pp 129ndash132 1962

[3] P J Olver Application of Lie groups to Differential EquationsSpringer New York NY USA 1993

[4] A H Kara and FMMahomed ldquoRelationship between symme-tries and conservation lawsrdquo International Journal ofTheoreticalPhysics vol 39 no 1 pp 23ndash40 2000

[5] S C Anco and G Bluman ldquoDirect construction method forconservation laws of partial differential equations I Examplesof conservation law classificationsrdquo European Journal of AppliedMathematics vol 13 no 5 pp 545ndash566 2002

[6] A H Kara and FMMahomed ldquoNoether-type symmetries andconservation laws via partial Lagrangiansrdquo Nonlinear Dynam-ics vol 45 no 3-4 pp 367ndash383 2006

[7] N H Ibragimov ldquoA new conservation theoremrdquo Journal ofMathematical Analysis and Applications vol 333 no 1 pp 311ndash328 2007

[8] R Naz F M Mahomed and D P Mason ldquoComparison ofdifferent approaches to conservation laws for some partialdifferential equations in fluid mechanicsrdquo Applied Mathematicsand Computation vol 205 no 1 pp 212ndash230 2008

[9] A F Cheviakov ldquoComputation of fluxes of conservation lawsrdquoJournal of EngineeringMathematics vol 66 no 1ndash3 pp 153ndash1732010

[10] T Wolf ldquoA comparison of four approaches to the calculation ofconservation lawsrdquo European Journal of Applied Mathematicsvol 13 no 2 pp 129ndash152 2002

[11] J K Hunter and R Saxton ldquoDynamics of director fieldsrdquo SIAMJournal on Applied Mathematics vol 51 no 6 pp 1498ndash15211991

[12] A G Johnpillai and C M Khalique ldquoSymmetry reductionsexact solutions and conservation laws of a modified Hunter-Saxton equationrdquo Abstract and Applied Analysis vol 2013Article ID 204746 5 pages 2013

[13] B Khesin and G Misiołek ldquoEuler equations on homogeneousspaces and Virasoro orbitsrdquo Advances in Mathematics vol 176no 1 pp 116ndash144 2003

[14] J K Hunter and Y X Zheng ldquoOn a completely integrable non-linear hyperbolic variational equationrdquo Physica D NonlinearPhenomena vol 79 no 2ndash4 pp 361ndash386 1994

[15] M Nadjafikhah and F Ahangari ldquoSymmetry analysis andconservation laws for the Hunter-Saxton equationrdquo Communi-cations in Theoretical Physics vol 59 no 3 pp 335ndash348 2013

[16] A Sjoberg ldquoDouble reduction of PDEs from the association ofsymmetries with conservation laws with applicationsrdquo AppliedMathematics and Computation vol 184 no 2 pp 608ndash6162007

[17] K R Adem andCM Khalique ldquoExact solutions and conserva-tion laws of a (2+1)-dimensional nonlinear KP-BBM equationrdquoAbstract and Applied Analysis vol 2013 Article ID 791863 5pages 2013

Submit your manuscripts athttpwwwhindawicom

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MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 3

Definition 3 We say that (2) is quasi-self-adjoint if the adjointequation (12) becomes equivalent to (2) after the substitution119908 = 120601(119906) 120601(119906) = 0

32 The Multiplier Method Amultiplier Λ120572(119909 119906 119906

119909 ) has

the property that

Λ120572119864120572= 119863119894119879119894 (17)

holds identically Here we will consider multipliers of thirdorder that is Λ

120572= Λ

120572(119909 119905 119906 119906

119909 119906119909119909 119906119909119909119909

) The righthand side of (17) is a divergence expressionThe determiningequation for the multiplier Λ

120572is

120575 (Λ120572119864120572)

120575119906120572

= 0 (18)

Once the multipliers are obtained the conserved vectors arecalculated via a homotopy formula [5 17] All the multiplierscan be calculated with the aid of (18) for which the equationcan be expressed as a local conservation law [9]

33 Double Reduction Method Let 119883 be any Lie-point sym-metry and 119879119894 are the components of conserved vector If 119883and 119879 satisfy

119883(119879119894) + 119879119894119863119895(120585119895) minus 119879119895119863119895(120585119894) = 0 119894 = 1 2 (19)

then 119883 is associated with 119879 We define a nonlocal variable Vby 119879119905 = V

119909 119879119909 = minusV

119905 Taking the similarity variables 119903 119904 120603

with the generator119883 = 120597120597119904 we have in similarity variables

119879119903= V119904 119879

119904= minusV119903 (20)

so that the conservation law is rewritten as

119863119903119879119903+ 119863119904119879119904= 0 (21)

Using the chain rule we have

119863119909= 119863119909(119904)119863119904+ 119863119909(119903)119863119903 119863

119905= 119863119905(119904)119863119904+ 119863119905(119903)119863119903

(22)

so that

V119909= V119904119863119909(119904) + V

119903119863119909(119903) V

119905= V119904119863119905(119904) + V

119903119863119905(119903) (23)

and so

119879119905= 119879119903119863119909(119904) minus 119879

119904119863119909(119903) 119879

119909= 119879119903119863119909(119904) minus 119879

119904119863119905(119903)

(24)

Using the above linear algebraical system we can get

119879119904=

119879119905119863119905(119904) + 119879

119909119863119909(119904)

119863119905(119903)119863119909(119904) minus 119863

119909(119903)119863119905(119904)

(25)

119879119903=

119879119905119863119905(119903) + 119879

119909119863119909(119903)

119863119905(119903)119863119909(119904) minus 119863

119909(119903)119863119905(119904)

(26)

The components 119879119909 119879119905 depend on (119905 119909 119906 119906(1) 119906(2)

119906(119902minus1)

) which means that 119879119904 119879119903 depend on (119904 119903 120603 120603

119903

120603119903119903 120603

119903(119902minus1)) for solutions invariant under 119883 Therefore

(21) becomes (120597119879119904120597119904) + 119863119903119879119903= 0

For 119879 associated with 119883 we have 119883119879119903 = 0 and 119883119879119904 = 0Thus 119879119903 and 119879119903 are invariant under 119883 This means 120597119879119904120597119904 =0 and 120597119879

119903120597119904 = 0 so that 119879119903(119903 120603 120603

119903 120603119903119903 120603

119903(119902minus1)) = 119896

where 119896 is constantEquation (2) of order 119902 with two independent variables

which admits a symmetry 119883 that is associated with aconserved vector 119879 is reduced to an ODE of order 119902 minus 1namely 119879119903 = 119896 where 119879

119903 is given by (26) for solutionsinvariant under119883

4 Main Results

Firstly we use the nonlocal conservation method given byIbragimov Equation (1) admits the following three Lie-pointsymmetry generators [12]

1198831=

120597

120597119909

1198832=

120597

120597119905

1198833= 119905

120597

120597119905

minus 119906

120597

120597119906

(27)

Equation (1) does not have the usual Lagrangian TheLagrangian for (1) is

119871 = 119908 (119906119905minus 2119906119909119906119909119909minus 119906119909119909119909

119906 minus 119906119909119909119905) = 0 (28)

The adjoint equation for (1) is

119864lowast(119905 119909 119906 119908 119908

119909119909119909119909)

=

120575

120575119906

[119908 (119906119905minus 2119906119909119906119909119909minus 119906119909119909119909

119906 minus 119906119909119909119905)]

(29)

and we can get the adjoint equation

119864lowast= 119908119909119906119909119909minus 119908119905+ 119908119909119909119906119909+ 119908119909119909119909

119906 + 119908119909119909119905

= 0 (30)

where 119908 is the adjoint variable Let us investigate the quasi-self-adjointness of (1) We make the ansatz of 119908 = 120601(119906)Taking into account (29) of 119864lowast and using (16) together withits consequences 119908 = 120601(119906) 119908

119905= 1206011015840119906119905 119908119909= 1206011015840119906119909 119908119909119909

=

120601101584010158401199062

119909+1206011015840119906119909119909119908119909119909119909

= 1206011015840101584010158401199063

119909+1206011015840119906119909119909119909

+312060110158401015840119906119909119906119909119909 and 119908

119909119909119905=

1206011015840101584010158401199061199051199062

119909+212060110158401015840119906119909119906119909119905+12060110158401015840119906119905119906119909119909+1206011015840119906119909119909119905

we rewrite (30) in thefollowing form

212060110158401015840119906119909119906119909119909minus 1206011015840119906119905+ 120601101584010158401199063

119909+ 1199061206011015840101584010158401199063

119909+ 3119906120601

10158401015840119906119909119906119909119909

+ 1199061206011015840119906119909119909119909

+ 1206011015840101584010158401199061199051199062

119909+ 212060110158401015840119906119909119906119909119905

+ 12060110158401015840119906119905119906119909119909+ 1206011015840119906119909119909119905

= 120582 (119906119905minus 2119906119909119906119909119909minus 119906119909119909119909

119906 minus 119906119909119909119905)

(31)

Equation (31) should be satisfied identically in all vari-ables 119906

119905 119906119909 119906119909119909 Comparing the coefficients of 119906

119905in both

sides of (31) we can easily obtain 120582 = minus1206011015840 Then we equate all

coefficients of linear and nonlinear mixed derivatives termsand get 120601(119906) = 119888

1119906 + 1198882

4 Advances in Mathematical Physics

The conserved components of (1) associated with asymmetry can be obtained from (14) as follows

119879119905= 120585119905119871 +119882(

120597119871

120597119906119905

+ 1198632

119909

120597119871

120597119906119909119909119905

) + 119863119909(119882)(minus119863

119909

120597119871

120597119906119909119909119905

)

+ 1198632

119909(119882)(

120597119871

120597119906119909119909119905

)

119879119909= 120585119909119871 +119882(

120597119871

120597119906119909

minus 119863119909

120597119871

120597119906119909119909

+ 1198632

119909

120597119871

120597119906119909119909119909

+ 119863119909119863119905

120597119871

120597119906119905119909119909

)

+ 119863119909(119882)(

120597119871

120597119906119909119909

minus 119863119909

120597119871

120597119906119909119909119909

minus 119863119905

120597119871

120597119906119905119909119909

)

+ 119863119905(119882)(minus119863

119909

120597119871

120597119906119905119909119909

) + 119863119909119863119905(119882)(

120597119871

120597119906119909119909119905

)

+ 1198632

119909(119882)(

120597119871

120597119906119909119909119909

)

(32)

where119882 is Lie characteristic function According to (31) wecan determine 119908 at two cases 119888

1= 1 1198882= 0 and 119888

1= 0 1198882= 1

has an infinite number of solutions The conservation lawsassociated with the generators (27) are below Firstly we take119908 = 119906

Case 1 Now let us make calculations for the operator 1198831=

120597120597119909 in detail For this operator the infinitesimals are 120585119909 =1 120585119905 = 0 and 120578 = 0 and we get 119882 = minus119906

119909and the

corresponding conserved vector of (1) as

119879119909

1= minus119906119905119906

119879119905

1= 119906119909119906

(33)

It is readily seen that in this case we obtain null conservedvectors by the definition of conservation laws

Case 2 In this case for the generator1198832= 120597120597119905 (120585119909 = 0 120585119905 =

1 and 120578 = 0) we calculate 119882 = minus119906119905and the conserved

quantities of (1) as

119879119909

2= 2119906119905119906119909119909119906 minus 119906119905119905119906119909+ 1199061199051199091199091199062+ 119906119905119905119909119906

119879119905

2= minus 2119906

119909119906119909119909119906 minus 1199062119906119909119909119909

+ 119906119905119906119909119909minus 119906119905119909119906119909

(34)

The divergence condition becomes

119863119905119879119905+ 119863119909119879119909= minus1199062

119905119909+ 119906119905119905119909119909

119906 minus 119906119905119905119909119906119909+ 119906119905119909119909119906119905 (35)

We observe that extra terms emerge By some adjustmentsthese terms can be absorbed as

119863119905119879119905+ 119863119909119879119909= 119863119905(minus119906119905119909119906119909+ 119906119905119909119909119906) (36)

into the conservation law Taking these terms across andincluding them into the conserved flows we get

119879119909

2= 2119906119905119906119909119909119906 minus 119906119905119905119906119909+ 1199061199051199091199091199062+ 119906119905119905119909119906

119879119905

2= minus2119906

119909119906119909119909119906 minus 1199062119906119909119909119909

+ 119906119905119906119909119909minus 119906119905119909119909119906

(37)

The modified conserved quantities are now labeled 119879119894 where

119863119905(119879119905) + 119863

119909(119879119909) = 0 modulo the equation It is readily

seen that in this case we obtain null conserved vectors by thedefinition of conservation laws

Case 3 Let us find the conservation law provided by 1198833=

119905(120597120597119905) minus 119906(120597120597119906) (120585119909 = 0 120585119905 = 119905 and 120578 = minus119906) In this case wehave 119882 = minus119906 minus 119905119906

119905and (32) yield the conservation laws (9)

with

119879119909

3= 31199062119906119909119909+ 3119906119906

119905119909minus 3119906119905119906119909+ 2119905119906119906

119905119906119909119909

minus 119905119906119909119906119905119905+ 1199051199062119906119905119909119909

+ 119905119906119906119905119905119909

119879119905

3= minus2119905119906119906

119909119906119909119909minus 1199051199062119906119909119909119909

minus 1199062+ 2119906119906

119909119909

+ 119905119906119905119906119909119909+ 1199062

119909+ 119905119906119909119906119905119909

(38)

The divergence of (38) is

119863119905119879119905+ 119863119909119879119909= 119863119909(119905119906119905119906119905119909+ 3119906119906

119905119909+ 119905119906119906119909119909119905) (39)

After some adjustments the nontrivial conserved quantitiesare as follows

119879119909

3= 31199062119906119909119909minus 3119906119905119906119909+ 2119905119906119906

119905119906119909119909

minus 119905119906119909119906119905119905+ 1199051199062119906119905119909119909

minus 119905119906119905119906119905119909

119879119905

3= minus2119905119906119906

119909119906119909119909minus 1199051199062119906119909119909119909

minus 1199062+ 2119906119906

119909119909

+ 119905119906119905119906119909119909+ 1199062

119909+ 119905119906119909119906119905119909

(40)

For the second case 119908(119909 119905) = 1 the corresponding con-servation laws are as follows

Case 4 For the generator 1198831= 120597120597119909 and Lie characteristic

function119882 = minus119906119909 we get the following conserved vectors

119879119909

4= 119906119905

119879119905

4= minus119906119909

(41)

Again like in Case 1 we obtain the null conserved vectors

Case 5 In this case for the generator 1198832= 120597120597119905 (120585119909 = 0

120585119905= 1 and 120578 = 0) we calculate119882 = minus119906

119905and the conserved

quantities of (1) as

119879119909

5= 119906119905119906119909119909+ 119906119905119909119906119909+ 119906119905119909119909119906 + 119906119905119905119909

119879119905

5= minus 2119906

119909119906119909119909minus 119906119906119909119909119909

(42)

After adjustment according to divergence we get modifiedconserved vectors

119879119909

5= 119906119905119906119909119909+ 119906119905119909119906119909+ 119906119905119909119909119906 + 119906119905119905119909

119879119905

5= minus 2119906

119909119906119909119909minus 119906119906119909119909119909

minus 119906119909119909119905

(43)

Again like in Case 2 we obtain the null conserved vectors

Advances in Mathematical Physics 5

Case 6 Lastly we consider the generator 1198833= 119905(120597120597119905) minus

119906(120597120597119906) where 120585119909 = 0 120585119905 = 119905 and 120578 = minus119906 In this case wehave 119882 = minus119906 minus 119905119906

119905and (32) yield the conservation laws (9)

with

119879119909

6= 2119906119906

119909119909+ 119905119906119909119909119906119905+ 1199062

119909+ 119905119906119909119906119905119909

+ 119906119905119906119905119909119909

+ 2119906119905119909+ 119905119906119905119905119909

119879119905

6= minus2119905119906

119909119906119909119909minus 119906119905119906119909119909119909

minus 119906 + 119906119909119909

(44)

We calculate the divergence

119863119905119879119905+ 119863119909119879119909= 119863119905(119905119906119909119909119905

+ 119906119909119909) (45)

Following the same line we find that the modified nontrivialconserved vectors are

119879119909

6= 2119906119906

119909119909+ 119905119906119909119909119906119905+ 1199062

119909+ 119905119906119909119906119905119909

+ 119905119906119906119905119909119909

+ 2119906119905119909+ 119905119906119905119905119909

119879119905

6= minus2119905119906

119909119906119909119909minus 119905119906119906119909119909119909

minus 119906 minus 119905119906119909119909119905

(46)

Now we will derive the conservation laws of the MHSequation by themultipliermethodThe third ordermultiplierfor (1) is Λ(119909 119905 119906 119906

119909 119906119909119909 119906119909119909119909

119906119909119909119905) and the corresponding

determining equation is

120575

120575119906

[Λ (119906119905minus 2119906119909119906119909119909minus 119906119909119909119909

119906 minus 119906119909119909119905)] = 0 (47)

Expanding and then separating (47) with respect to differentcombinations of derivatives of 119906 yields the following overde-termined system for the multipliers

Λ119906119906= 0 Λ

119905= 0 Λ

119909= 0 Λ

119906119909

= 0

Λ119906119909119909

= 0 Λ119906119909119909119909

= 0 Λ119906119905119909119909

= 0

(48)

The solution of system (48) can be expressed as

Λ = 1198881119906 + 1198882 (49)

where 1198881 1198882are constants Corresponding to the above

multiplier we have the following conserved vectors of (49)

119879119909

1= minus 119906

2119906119909119909minus

119906119905119909119906

2

+

119906119909119906119905

2

119879119905

1=

1

2

1199062minus

1

2

119906119909119909119906

119879119909

2= minus 119906

119909119909119906 minus

1199062

119909

2

119879119905

2= 119906 minus 119906

119909119909

(50)

The multiplier approach gave two local conservation laws forthe MHS equation

Now we will derive the exact group-invariant solutionof (1) using the relationship between local conservation lawsand Lie-point symmetries Equation (1) admits the symmetrygenerators 119883

1= 120597120597119909 119883

2= 120597120597119905 associated with the

conservation law

119863119905(119906 minus 119906

119909119909) + 119863119909(minus119906119909119909119906 minus

1199062

119909

2

) = 0 (51)

We set 119883 = 1198831+ 1205721198832 Then the canonical coordinates of 119883

are 119904 = 119909 119903 = 120572119909 minus 119905 and 119906 Since 119879 = (119879119903 119879119904) is associated

with119883 we have to find the value of 119879119903 Using (26) we obtainthe following conserved vector

119879119903= 119896 = 120572

2119906119906119903119903+

1205722

2

1199062

119903 (52)

We can substitute the variables 119906119903= 119901 and 119906

119903119903= 119901(119889119901119889119906)

in (52) After using these variables (52) reduces to first orderordinary differential equation (ODE)

119901 (119906) = plusmn

radic119906 (2119896119906 + 11988811205722)

119906120572

(53)

We can solve (53) by separation of variables and the solutiongives rise to

119903 + 1198882

= plusmn

120572

2119896

radic21198961199062+ 12057221199061198881∓

1

8

times ((12057231198881ln(1

2

((12) 12057221198881+ 2119896119906)radic2

radic119896

+radic21198961199062+ 12057221199061198881)radic2) times (119896

32)

minus1

)

119903 = 120572119909 minus 119905

(54)

which constitutes the solution of the MHS equation

5 Conclusion

In this work we studied conservation laws symmetry reduc-tions and exact solutions of MHS equation Utilizing nonlo-cal conservation and multiplier method we constructed fourdistinct local conservation laws (see (40) (46) and (50)) It isclear that by using Ibragimovrsquos nonlocal conservationmethodone can obtain infinite nonlocal conservation laws Thenusing the double reduction method we reduced the MHSequation to second order ODE in the canonical variables (see(52)) Exact group-invariant solutions were constructed byintegrating the reduced ODE

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by Eskisehir Osmangazi UniversityScientific Research Projects (Grant no 2013-281)

6 Advances in Mathematical Physics

References

[1] E Noether ldquoInvariante Variationsproblemerdquo Nachrichten vonder koniglichen Gesellschaft der Wissenschaften zu Gottingenvol 2 pp 235ndash257 1918 English translation in TransportTheoryand Statistical Physics vol 1 no 3 pp 186ndash207 1971

[2] H Steudel ldquoUber die Zuordnung zwischen Invarianzeigen-schaften und Erhaltungssatzenrdquo Zeitschrift fur Naturforschungvol 17 pp 129ndash132 1962

[3] P J Olver Application of Lie groups to Differential EquationsSpringer New York NY USA 1993

[4] A H Kara and FMMahomed ldquoRelationship between symme-tries and conservation lawsrdquo International Journal ofTheoreticalPhysics vol 39 no 1 pp 23ndash40 2000

[5] S C Anco and G Bluman ldquoDirect construction method forconservation laws of partial differential equations I Examplesof conservation law classificationsrdquo European Journal of AppliedMathematics vol 13 no 5 pp 545ndash566 2002

[6] A H Kara and FMMahomed ldquoNoether-type symmetries andconservation laws via partial Lagrangiansrdquo Nonlinear Dynam-ics vol 45 no 3-4 pp 367ndash383 2006

[7] N H Ibragimov ldquoA new conservation theoremrdquo Journal ofMathematical Analysis and Applications vol 333 no 1 pp 311ndash328 2007

[8] R Naz F M Mahomed and D P Mason ldquoComparison ofdifferent approaches to conservation laws for some partialdifferential equations in fluid mechanicsrdquo Applied Mathematicsand Computation vol 205 no 1 pp 212ndash230 2008

[9] A F Cheviakov ldquoComputation of fluxes of conservation lawsrdquoJournal of EngineeringMathematics vol 66 no 1ndash3 pp 153ndash1732010

[10] T Wolf ldquoA comparison of four approaches to the calculation ofconservation lawsrdquo European Journal of Applied Mathematicsvol 13 no 2 pp 129ndash152 2002

[11] J K Hunter and R Saxton ldquoDynamics of director fieldsrdquo SIAMJournal on Applied Mathematics vol 51 no 6 pp 1498ndash15211991

[12] A G Johnpillai and C M Khalique ldquoSymmetry reductionsexact solutions and conservation laws of a modified Hunter-Saxton equationrdquo Abstract and Applied Analysis vol 2013Article ID 204746 5 pages 2013

[13] B Khesin and G Misiołek ldquoEuler equations on homogeneousspaces and Virasoro orbitsrdquo Advances in Mathematics vol 176no 1 pp 116ndash144 2003

[14] J K Hunter and Y X Zheng ldquoOn a completely integrable non-linear hyperbolic variational equationrdquo Physica D NonlinearPhenomena vol 79 no 2ndash4 pp 361ndash386 1994

[15] M Nadjafikhah and F Ahangari ldquoSymmetry analysis andconservation laws for the Hunter-Saxton equationrdquo Communi-cations in Theoretical Physics vol 59 no 3 pp 335ndash348 2013

[16] A Sjoberg ldquoDouble reduction of PDEs from the association ofsymmetries with conservation laws with applicationsrdquo AppliedMathematics and Computation vol 184 no 2 pp 608ndash6162007

[17] K R Adem andCM Khalique ldquoExact solutions and conserva-tion laws of a (2+1)-dimensional nonlinear KP-BBM equationrdquoAbstract and Applied Analysis vol 2013 Article ID 791863 5pages 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Advances in Mathematical Physics

The conserved components of (1) associated with asymmetry can be obtained from (14) as follows

119879119905= 120585119905119871 +119882(

120597119871

120597119906119905

+ 1198632

119909

120597119871

120597119906119909119909119905

) + 119863119909(119882)(minus119863

119909

120597119871

120597119906119909119909119905

)

+ 1198632

119909(119882)(

120597119871

120597119906119909119909119905

)

119879119909= 120585119909119871 +119882(

120597119871

120597119906119909

minus 119863119909

120597119871

120597119906119909119909

+ 1198632

119909

120597119871

120597119906119909119909119909

+ 119863119909119863119905

120597119871

120597119906119905119909119909

)

+ 119863119909(119882)(

120597119871

120597119906119909119909

minus 119863119909

120597119871

120597119906119909119909119909

minus 119863119905

120597119871

120597119906119905119909119909

)

+ 119863119905(119882)(minus119863

119909

120597119871

120597119906119905119909119909

) + 119863119909119863119905(119882)(

120597119871

120597119906119909119909119905

)

+ 1198632

119909(119882)(

120597119871

120597119906119909119909119909

)

(32)

where119882 is Lie characteristic function According to (31) wecan determine 119908 at two cases 119888

1= 1 1198882= 0 and 119888

1= 0 1198882= 1

has an infinite number of solutions The conservation lawsassociated with the generators (27) are below Firstly we take119908 = 119906

Case 1 Now let us make calculations for the operator 1198831=

120597120597119909 in detail For this operator the infinitesimals are 120585119909 =1 120585119905 = 0 and 120578 = 0 and we get 119882 = minus119906

119909and the

corresponding conserved vector of (1) as

119879119909

1= minus119906119905119906

119879119905

1= 119906119909119906

(33)

It is readily seen that in this case we obtain null conservedvectors by the definition of conservation laws

Case 2 In this case for the generator1198832= 120597120597119905 (120585119909 = 0 120585119905 =

1 and 120578 = 0) we calculate 119882 = minus119906119905and the conserved

quantities of (1) as

119879119909

2= 2119906119905119906119909119909119906 minus 119906119905119905119906119909+ 1199061199051199091199091199062+ 119906119905119905119909119906

119879119905

2= minus 2119906

119909119906119909119909119906 minus 1199062119906119909119909119909

+ 119906119905119906119909119909minus 119906119905119909119906119909

(34)

The divergence condition becomes

119863119905119879119905+ 119863119909119879119909= minus1199062

119905119909+ 119906119905119905119909119909

119906 minus 119906119905119905119909119906119909+ 119906119905119909119909119906119905 (35)

We observe that extra terms emerge By some adjustmentsthese terms can be absorbed as

119863119905119879119905+ 119863119909119879119909= 119863119905(minus119906119905119909119906119909+ 119906119905119909119909119906) (36)

into the conservation law Taking these terms across andincluding them into the conserved flows we get

119879119909

2= 2119906119905119906119909119909119906 minus 119906119905119905119906119909+ 1199061199051199091199091199062+ 119906119905119905119909119906

119879119905

2= minus2119906

119909119906119909119909119906 minus 1199062119906119909119909119909

+ 119906119905119906119909119909minus 119906119905119909119909119906

(37)

The modified conserved quantities are now labeled 119879119894 where

119863119905(119879119905) + 119863

119909(119879119909) = 0 modulo the equation It is readily

seen that in this case we obtain null conserved vectors by thedefinition of conservation laws

Case 3 Let us find the conservation law provided by 1198833=

119905(120597120597119905) minus 119906(120597120597119906) (120585119909 = 0 120585119905 = 119905 and 120578 = minus119906) In this case wehave 119882 = minus119906 minus 119905119906

119905and (32) yield the conservation laws (9)

with

119879119909

3= 31199062119906119909119909+ 3119906119906

119905119909minus 3119906119905119906119909+ 2119905119906119906

119905119906119909119909

minus 119905119906119909119906119905119905+ 1199051199062119906119905119909119909

+ 119905119906119906119905119905119909

119879119905

3= minus2119905119906119906

119909119906119909119909minus 1199051199062119906119909119909119909

minus 1199062+ 2119906119906

119909119909

+ 119905119906119905119906119909119909+ 1199062

119909+ 119905119906119909119906119905119909

(38)

The divergence of (38) is

119863119905119879119905+ 119863119909119879119909= 119863119909(119905119906119905119906119905119909+ 3119906119906

119905119909+ 119905119906119906119909119909119905) (39)

After some adjustments the nontrivial conserved quantitiesare as follows

119879119909

3= 31199062119906119909119909minus 3119906119905119906119909+ 2119905119906119906

119905119906119909119909

minus 119905119906119909119906119905119905+ 1199051199062119906119905119909119909

minus 119905119906119905119906119905119909

119879119905

3= minus2119905119906119906

119909119906119909119909minus 1199051199062119906119909119909119909

minus 1199062+ 2119906119906

119909119909

+ 119905119906119905119906119909119909+ 1199062

119909+ 119905119906119909119906119905119909

(40)

For the second case 119908(119909 119905) = 1 the corresponding con-servation laws are as follows

Case 4 For the generator 1198831= 120597120597119909 and Lie characteristic

function119882 = minus119906119909 we get the following conserved vectors

119879119909

4= 119906119905

119879119905

4= minus119906119909

(41)

Again like in Case 1 we obtain the null conserved vectors

Case 5 In this case for the generator 1198832= 120597120597119905 (120585119909 = 0

120585119905= 1 and 120578 = 0) we calculate119882 = minus119906

119905and the conserved

quantities of (1) as

119879119909

5= 119906119905119906119909119909+ 119906119905119909119906119909+ 119906119905119909119909119906 + 119906119905119905119909

119879119905

5= minus 2119906

119909119906119909119909minus 119906119906119909119909119909

(42)

After adjustment according to divergence we get modifiedconserved vectors

119879119909

5= 119906119905119906119909119909+ 119906119905119909119906119909+ 119906119905119909119909119906 + 119906119905119905119909

119879119905

5= minus 2119906

119909119906119909119909minus 119906119906119909119909119909

minus 119906119909119909119905

(43)

Again like in Case 2 we obtain the null conserved vectors

Advances in Mathematical Physics 5

Case 6 Lastly we consider the generator 1198833= 119905(120597120597119905) minus

119906(120597120597119906) where 120585119909 = 0 120585119905 = 119905 and 120578 = minus119906 In this case wehave 119882 = minus119906 minus 119905119906

119905and (32) yield the conservation laws (9)

with

119879119909

6= 2119906119906

119909119909+ 119905119906119909119909119906119905+ 1199062

119909+ 119905119906119909119906119905119909

+ 119906119905119906119905119909119909

+ 2119906119905119909+ 119905119906119905119905119909

119879119905

6= minus2119905119906

119909119906119909119909minus 119906119905119906119909119909119909

minus 119906 + 119906119909119909

(44)

We calculate the divergence

119863119905119879119905+ 119863119909119879119909= 119863119905(119905119906119909119909119905

+ 119906119909119909) (45)

Following the same line we find that the modified nontrivialconserved vectors are

119879119909

6= 2119906119906

119909119909+ 119905119906119909119909119906119905+ 1199062

119909+ 119905119906119909119906119905119909

+ 119905119906119906119905119909119909

+ 2119906119905119909+ 119905119906119905119905119909

119879119905

6= minus2119905119906

119909119906119909119909minus 119905119906119906119909119909119909

minus 119906 minus 119905119906119909119909119905

(46)

Now we will derive the conservation laws of the MHSequation by themultipliermethodThe third ordermultiplierfor (1) is Λ(119909 119905 119906 119906

119909 119906119909119909 119906119909119909119909

119906119909119909119905) and the corresponding

determining equation is

120575

120575119906

[Λ (119906119905minus 2119906119909119906119909119909minus 119906119909119909119909

119906 minus 119906119909119909119905)] = 0 (47)

Expanding and then separating (47) with respect to differentcombinations of derivatives of 119906 yields the following overde-termined system for the multipliers

Λ119906119906= 0 Λ

119905= 0 Λ

119909= 0 Λ

119906119909

= 0

Λ119906119909119909

= 0 Λ119906119909119909119909

= 0 Λ119906119905119909119909

= 0

(48)

The solution of system (48) can be expressed as

Λ = 1198881119906 + 1198882 (49)

where 1198881 1198882are constants Corresponding to the above

multiplier we have the following conserved vectors of (49)

119879119909

1= minus 119906

2119906119909119909minus

119906119905119909119906

2

+

119906119909119906119905

2

119879119905

1=

1

2

1199062minus

1

2

119906119909119909119906

119879119909

2= minus 119906

119909119909119906 minus

1199062

119909

2

119879119905

2= 119906 minus 119906

119909119909

(50)

The multiplier approach gave two local conservation laws forthe MHS equation

Now we will derive the exact group-invariant solutionof (1) using the relationship between local conservation lawsand Lie-point symmetries Equation (1) admits the symmetrygenerators 119883

1= 120597120597119909 119883

2= 120597120597119905 associated with the

conservation law

119863119905(119906 minus 119906

119909119909) + 119863119909(minus119906119909119909119906 minus

1199062

119909

2

) = 0 (51)

We set 119883 = 1198831+ 1205721198832 Then the canonical coordinates of 119883

are 119904 = 119909 119903 = 120572119909 minus 119905 and 119906 Since 119879 = (119879119903 119879119904) is associated

with119883 we have to find the value of 119879119903 Using (26) we obtainthe following conserved vector

119879119903= 119896 = 120572

2119906119906119903119903+

1205722

2

1199062

119903 (52)

We can substitute the variables 119906119903= 119901 and 119906

119903119903= 119901(119889119901119889119906)

in (52) After using these variables (52) reduces to first orderordinary differential equation (ODE)

119901 (119906) = plusmn

radic119906 (2119896119906 + 11988811205722)

119906120572

(53)

We can solve (53) by separation of variables and the solutiongives rise to

119903 + 1198882

= plusmn

120572

2119896

radic21198961199062+ 12057221199061198881∓

1

8

times ((12057231198881ln(1

2

((12) 12057221198881+ 2119896119906)radic2

radic119896

+radic21198961199062+ 12057221199061198881)radic2) times (119896

32)

minus1

)

119903 = 120572119909 minus 119905

(54)

which constitutes the solution of the MHS equation

5 Conclusion

In this work we studied conservation laws symmetry reduc-tions and exact solutions of MHS equation Utilizing nonlo-cal conservation and multiplier method we constructed fourdistinct local conservation laws (see (40) (46) and (50)) It isclear that by using Ibragimovrsquos nonlocal conservationmethodone can obtain infinite nonlocal conservation laws Thenusing the double reduction method we reduced the MHSequation to second order ODE in the canonical variables (see(52)) Exact group-invariant solutions were constructed byintegrating the reduced ODE

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by Eskisehir Osmangazi UniversityScientific Research Projects (Grant no 2013-281)

6 Advances in Mathematical Physics

References

[1] E Noether ldquoInvariante Variationsproblemerdquo Nachrichten vonder koniglichen Gesellschaft der Wissenschaften zu Gottingenvol 2 pp 235ndash257 1918 English translation in TransportTheoryand Statistical Physics vol 1 no 3 pp 186ndash207 1971

[2] H Steudel ldquoUber die Zuordnung zwischen Invarianzeigen-schaften und Erhaltungssatzenrdquo Zeitschrift fur Naturforschungvol 17 pp 129ndash132 1962

[3] P J Olver Application of Lie groups to Differential EquationsSpringer New York NY USA 1993

[4] A H Kara and FMMahomed ldquoRelationship between symme-tries and conservation lawsrdquo International Journal ofTheoreticalPhysics vol 39 no 1 pp 23ndash40 2000

[5] S C Anco and G Bluman ldquoDirect construction method forconservation laws of partial differential equations I Examplesof conservation law classificationsrdquo European Journal of AppliedMathematics vol 13 no 5 pp 545ndash566 2002

[6] A H Kara and FMMahomed ldquoNoether-type symmetries andconservation laws via partial Lagrangiansrdquo Nonlinear Dynam-ics vol 45 no 3-4 pp 367ndash383 2006

[7] N H Ibragimov ldquoA new conservation theoremrdquo Journal ofMathematical Analysis and Applications vol 333 no 1 pp 311ndash328 2007

[8] R Naz F M Mahomed and D P Mason ldquoComparison ofdifferent approaches to conservation laws for some partialdifferential equations in fluid mechanicsrdquo Applied Mathematicsand Computation vol 205 no 1 pp 212ndash230 2008

[9] A F Cheviakov ldquoComputation of fluxes of conservation lawsrdquoJournal of EngineeringMathematics vol 66 no 1ndash3 pp 153ndash1732010

[10] T Wolf ldquoA comparison of four approaches to the calculation ofconservation lawsrdquo European Journal of Applied Mathematicsvol 13 no 2 pp 129ndash152 2002

[11] J K Hunter and R Saxton ldquoDynamics of director fieldsrdquo SIAMJournal on Applied Mathematics vol 51 no 6 pp 1498ndash15211991

[12] A G Johnpillai and C M Khalique ldquoSymmetry reductionsexact solutions and conservation laws of a modified Hunter-Saxton equationrdquo Abstract and Applied Analysis vol 2013Article ID 204746 5 pages 2013

[13] B Khesin and G Misiołek ldquoEuler equations on homogeneousspaces and Virasoro orbitsrdquo Advances in Mathematics vol 176no 1 pp 116ndash144 2003

[14] J K Hunter and Y X Zheng ldquoOn a completely integrable non-linear hyperbolic variational equationrdquo Physica D NonlinearPhenomena vol 79 no 2ndash4 pp 361ndash386 1994

[15] M Nadjafikhah and F Ahangari ldquoSymmetry analysis andconservation laws for the Hunter-Saxton equationrdquo Communi-cations in Theoretical Physics vol 59 no 3 pp 335ndash348 2013

[16] A Sjoberg ldquoDouble reduction of PDEs from the association ofsymmetries with conservation laws with applicationsrdquo AppliedMathematics and Computation vol 184 no 2 pp 608ndash6162007

[17] K R Adem andCM Khalique ldquoExact solutions and conserva-tion laws of a (2+1)-dimensional nonlinear KP-BBM equationrdquoAbstract and Applied Analysis vol 2013 Article ID 791863 5pages 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 5

Case 6 Lastly we consider the generator 1198833= 119905(120597120597119905) minus

119906(120597120597119906) where 120585119909 = 0 120585119905 = 119905 and 120578 = minus119906 In this case wehave 119882 = minus119906 minus 119905119906

119905and (32) yield the conservation laws (9)

with

119879119909

6= 2119906119906

119909119909+ 119905119906119909119909119906119905+ 1199062

119909+ 119905119906119909119906119905119909

+ 119906119905119906119905119909119909

+ 2119906119905119909+ 119905119906119905119905119909

119879119905

6= minus2119905119906

119909119906119909119909minus 119906119905119906119909119909119909

minus 119906 + 119906119909119909

(44)

We calculate the divergence

119863119905119879119905+ 119863119909119879119909= 119863119905(119905119906119909119909119905

+ 119906119909119909) (45)

Following the same line we find that the modified nontrivialconserved vectors are

119879119909

6= 2119906119906

119909119909+ 119905119906119909119909119906119905+ 1199062

119909+ 119905119906119909119906119905119909

+ 119905119906119906119905119909119909

+ 2119906119905119909+ 119905119906119905119905119909

119879119905

6= minus2119905119906

119909119906119909119909minus 119905119906119906119909119909119909

minus 119906 minus 119905119906119909119909119905

(46)

Now we will derive the conservation laws of the MHSequation by themultipliermethodThe third ordermultiplierfor (1) is Λ(119909 119905 119906 119906

119909 119906119909119909 119906119909119909119909

119906119909119909119905) and the corresponding

determining equation is

120575

120575119906

[Λ (119906119905minus 2119906119909119906119909119909minus 119906119909119909119909

119906 minus 119906119909119909119905)] = 0 (47)

Expanding and then separating (47) with respect to differentcombinations of derivatives of 119906 yields the following overde-termined system for the multipliers

Λ119906119906= 0 Λ

119905= 0 Λ

119909= 0 Λ

119906119909

= 0

Λ119906119909119909

= 0 Λ119906119909119909119909

= 0 Λ119906119905119909119909

= 0

(48)

The solution of system (48) can be expressed as

Λ = 1198881119906 + 1198882 (49)

where 1198881 1198882are constants Corresponding to the above

multiplier we have the following conserved vectors of (49)

119879119909

1= minus 119906

2119906119909119909minus

119906119905119909119906

2

+

119906119909119906119905

2

119879119905

1=

1

2

1199062minus

1

2

119906119909119909119906

119879119909

2= minus 119906

119909119909119906 minus

1199062

119909

2

119879119905

2= 119906 minus 119906

119909119909

(50)

The multiplier approach gave two local conservation laws forthe MHS equation

Now we will derive the exact group-invariant solutionof (1) using the relationship between local conservation lawsand Lie-point symmetries Equation (1) admits the symmetrygenerators 119883

1= 120597120597119909 119883

2= 120597120597119905 associated with the

conservation law

119863119905(119906 minus 119906

119909119909) + 119863119909(minus119906119909119909119906 minus

1199062

119909

2

) = 0 (51)

We set 119883 = 1198831+ 1205721198832 Then the canonical coordinates of 119883

are 119904 = 119909 119903 = 120572119909 minus 119905 and 119906 Since 119879 = (119879119903 119879119904) is associated

with119883 we have to find the value of 119879119903 Using (26) we obtainthe following conserved vector

119879119903= 119896 = 120572

2119906119906119903119903+

1205722

2

1199062

119903 (52)

We can substitute the variables 119906119903= 119901 and 119906

119903119903= 119901(119889119901119889119906)

in (52) After using these variables (52) reduces to first orderordinary differential equation (ODE)

119901 (119906) = plusmn

radic119906 (2119896119906 + 11988811205722)

119906120572

(53)

We can solve (53) by separation of variables and the solutiongives rise to

119903 + 1198882

= plusmn

120572

2119896

radic21198961199062+ 12057221199061198881∓

1

8

times ((12057231198881ln(1

2

((12) 12057221198881+ 2119896119906)radic2

radic119896

+radic21198961199062+ 12057221199061198881)radic2) times (119896

32)

minus1

)

119903 = 120572119909 minus 119905

(54)

which constitutes the solution of the MHS equation

5 Conclusion

In this work we studied conservation laws symmetry reduc-tions and exact solutions of MHS equation Utilizing nonlo-cal conservation and multiplier method we constructed fourdistinct local conservation laws (see (40) (46) and (50)) It isclear that by using Ibragimovrsquos nonlocal conservationmethodone can obtain infinite nonlocal conservation laws Thenusing the double reduction method we reduced the MHSequation to second order ODE in the canonical variables (see(52)) Exact group-invariant solutions were constructed byintegrating the reduced ODE

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by Eskisehir Osmangazi UniversityScientific Research Projects (Grant no 2013-281)

6 Advances in Mathematical Physics

References

[1] E Noether ldquoInvariante Variationsproblemerdquo Nachrichten vonder koniglichen Gesellschaft der Wissenschaften zu Gottingenvol 2 pp 235ndash257 1918 English translation in TransportTheoryand Statistical Physics vol 1 no 3 pp 186ndash207 1971

[2] H Steudel ldquoUber die Zuordnung zwischen Invarianzeigen-schaften und Erhaltungssatzenrdquo Zeitschrift fur Naturforschungvol 17 pp 129ndash132 1962

[3] P J Olver Application of Lie groups to Differential EquationsSpringer New York NY USA 1993

[4] A H Kara and FMMahomed ldquoRelationship between symme-tries and conservation lawsrdquo International Journal ofTheoreticalPhysics vol 39 no 1 pp 23ndash40 2000

[5] S C Anco and G Bluman ldquoDirect construction method forconservation laws of partial differential equations I Examplesof conservation law classificationsrdquo European Journal of AppliedMathematics vol 13 no 5 pp 545ndash566 2002

[6] A H Kara and FMMahomed ldquoNoether-type symmetries andconservation laws via partial Lagrangiansrdquo Nonlinear Dynam-ics vol 45 no 3-4 pp 367ndash383 2006

[7] N H Ibragimov ldquoA new conservation theoremrdquo Journal ofMathematical Analysis and Applications vol 333 no 1 pp 311ndash328 2007

[8] R Naz F M Mahomed and D P Mason ldquoComparison ofdifferent approaches to conservation laws for some partialdifferential equations in fluid mechanicsrdquo Applied Mathematicsand Computation vol 205 no 1 pp 212ndash230 2008

[9] A F Cheviakov ldquoComputation of fluxes of conservation lawsrdquoJournal of EngineeringMathematics vol 66 no 1ndash3 pp 153ndash1732010

[10] T Wolf ldquoA comparison of four approaches to the calculation ofconservation lawsrdquo European Journal of Applied Mathematicsvol 13 no 2 pp 129ndash152 2002

[11] J K Hunter and R Saxton ldquoDynamics of director fieldsrdquo SIAMJournal on Applied Mathematics vol 51 no 6 pp 1498ndash15211991

[12] A G Johnpillai and C M Khalique ldquoSymmetry reductionsexact solutions and conservation laws of a modified Hunter-Saxton equationrdquo Abstract and Applied Analysis vol 2013Article ID 204746 5 pages 2013

[13] B Khesin and G Misiołek ldquoEuler equations on homogeneousspaces and Virasoro orbitsrdquo Advances in Mathematics vol 176no 1 pp 116ndash144 2003

[14] J K Hunter and Y X Zheng ldquoOn a completely integrable non-linear hyperbolic variational equationrdquo Physica D NonlinearPhenomena vol 79 no 2ndash4 pp 361ndash386 1994

[15] M Nadjafikhah and F Ahangari ldquoSymmetry analysis andconservation laws for the Hunter-Saxton equationrdquo Communi-cations in Theoretical Physics vol 59 no 3 pp 335ndash348 2013

[16] A Sjoberg ldquoDouble reduction of PDEs from the association ofsymmetries with conservation laws with applicationsrdquo AppliedMathematics and Computation vol 184 no 2 pp 608ndash6162007

[17] K R Adem andCM Khalique ldquoExact solutions and conserva-tion laws of a (2+1)-dimensional nonlinear KP-BBM equationrdquoAbstract and Applied Analysis vol 2013 Article ID 791863 5pages 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Advances in Mathematical Physics

References

[1] E Noether ldquoInvariante Variationsproblemerdquo Nachrichten vonder koniglichen Gesellschaft der Wissenschaften zu Gottingenvol 2 pp 235ndash257 1918 English translation in TransportTheoryand Statistical Physics vol 1 no 3 pp 186ndash207 1971

[2] H Steudel ldquoUber die Zuordnung zwischen Invarianzeigen-schaften und Erhaltungssatzenrdquo Zeitschrift fur Naturforschungvol 17 pp 129ndash132 1962

[3] P J Olver Application of Lie groups to Differential EquationsSpringer New York NY USA 1993

[4] A H Kara and FMMahomed ldquoRelationship between symme-tries and conservation lawsrdquo International Journal ofTheoreticalPhysics vol 39 no 1 pp 23ndash40 2000

[5] S C Anco and G Bluman ldquoDirect construction method forconservation laws of partial differential equations I Examplesof conservation law classificationsrdquo European Journal of AppliedMathematics vol 13 no 5 pp 545ndash566 2002

[6] A H Kara and FMMahomed ldquoNoether-type symmetries andconservation laws via partial Lagrangiansrdquo Nonlinear Dynam-ics vol 45 no 3-4 pp 367ndash383 2006

[7] N H Ibragimov ldquoA new conservation theoremrdquo Journal ofMathematical Analysis and Applications vol 333 no 1 pp 311ndash328 2007

[8] R Naz F M Mahomed and D P Mason ldquoComparison ofdifferent approaches to conservation laws for some partialdifferential equations in fluid mechanicsrdquo Applied Mathematicsand Computation vol 205 no 1 pp 212ndash230 2008

[9] A F Cheviakov ldquoComputation of fluxes of conservation lawsrdquoJournal of EngineeringMathematics vol 66 no 1ndash3 pp 153ndash1732010

[10] T Wolf ldquoA comparison of four approaches to the calculation ofconservation lawsrdquo European Journal of Applied Mathematicsvol 13 no 2 pp 129ndash152 2002

[11] J K Hunter and R Saxton ldquoDynamics of director fieldsrdquo SIAMJournal on Applied Mathematics vol 51 no 6 pp 1498ndash15211991

[12] A G Johnpillai and C M Khalique ldquoSymmetry reductionsexact solutions and conservation laws of a modified Hunter-Saxton equationrdquo Abstract and Applied Analysis vol 2013Article ID 204746 5 pages 2013

[13] B Khesin and G Misiołek ldquoEuler equations on homogeneousspaces and Virasoro orbitsrdquo Advances in Mathematics vol 176no 1 pp 116ndash144 2003

[14] J K Hunter and Y X Zheng ldquoOn a completely integrable non-linear hyperbolic variational equationrdquo Physica D NonlinearPhenomena vol 79 no 2ndash4 pp 361ndash386 1994

[15] M Nadjafikhah and F Ahangari ldquoSymmetry analysis andconservation laws for the Hunter-Saxton equationrdquo Communi-cations in Theoretical Physics vol 59 no 3 pp 335ndash348 2013

[16] A Sjoberg ldquoDouble reduction of PDEs from the association ofsymmetries with conservation laws with applicationsrdquo AppliedMathematics and Computation vol 184 no 2 pp 608ndash6162007

[17] K R Adem andCM Khalique ldquoExact solutions and conserva-tion laws of a (2+1)-dimensional nonlinear KP-BBM equationrdquoAbstract and Applied Analysis vol 2013 Article ID 791863 5pages 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of