Research ArticleSymmetry Analysis and Conservation Laws of a GeneralizedTwo-Dimensional Nonlinear KP-MEW Equation

Khadijo Rashid Adem and Chaudry Masood Khalique

International Institute for Symmetry Analysis and Mathematical Modelling Department of Mathematical SciencesNorth-West University Mafikeng Campus Private Bag X 2046 Mmabatho 2735 South Africa

Correspondence should be addressed to Chaudry Masood Khalique masoodkhaliquenwuacza

Received 2 August 2014 Accepted 20 September 2014

Academic Editor Hossein Jafari

Copyright copy 2015 K R Adem and C M Khalique This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Lie symmetry analysis is performed on a generalized two-dimensional nonlinear Kadomtsev-Petviashvili-modified equal widthequation The symmetries and adjoint representations for this equation are given and an optimal system of one-dimensionalsubalgebras is derived The similarity reductions and exact solutions with the aid of (1198661015840119866)-expansion method are obtained basedon the optimal systems of one-dimensional subalgebras Finally conservation laws are constructed by using the multiplier method

1 Introduction

Nonlinear evolution equations (NLEEs) have been widelyused to describe natural phenomena of science and engineer-ing Therefore it is very important to find exact solutionsof NLEEs However this is not an easy task During thepast few decades various integration techniques have beendeveloped by the researchers to solve these NLEEs Someof the well-known techniques used in the literature are theinverse scattering transform method [1] the homogeneousbalance method [2] the Backlund transformation [3] theWeierstrass elliptic function expansion method [4] the Dar-boux transformation [5] the ansatz method [6 7] Hirotarsquosbilinear method [8] the (119866

1015840119866)-expansion method [9] theJacobi elliptic function expansionmethod [10 11] the variableseparation approach [12] the sine-cosine method [13] thetrifunctionmethod [14 15] the F-expansionmethod [16] theexp-functionmethod [17] themultiple exp-functionmethod[18] and the Lie symmetry method [19ndash25]

The purpose of this paper is to study one such NLEEnamely the generalized two-dimensional nonlinear Kad-omtsev-Petviashvili-modified equal width (KP-MEW) equa-tion [26] that is given by

(119906119905+ 120572 (119906

119899)119909+ 120573119906119909119909119905

)119909+ 120574119906119910119910

= 0 (1)

Here in (1) 120572 120573 120574 and 119899 gt 1 are real valued constantsThe solutions of (1) have been studied in various aspectsSee for example the recent papers [26ndash28] Wazwaz [26]used the tanh method and the sine-cosine method forfinding solitary waves and periodic solutions Saha [27] usedthe theory of bifurcations of planar dynamical systems toprove the existence of smooth and nonsmooth travellingwave solutions Wei et al [28] used the qualitative theoryof differential equations and obtained peakon compactoncuspons loop soliton solutions and smooth soliton solutions

In this paper we obtain symmetry reductions of (1) usingLie group analysis [19ndash24] and based on the optimal systemsof one-dimensional subalgebras Furthermore the (119866

1015840119866)-

expansion method is employed to obtain some exact solu-tions of (1) In addition to this conservation laws will bederived for (1) using the multiplier method [29]

2 Symmetry Reductions and ExactSolutions of (1)

The vector field of the form

119883 = 1205851 120597

120597119909+ 1205852 120597

120597119910+ 1205853 120597

120597119905+ 120578

120597

120597119906 (2)

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 805763 7 pageshttpdxdoiorg1011552015805763

2 Mathematical Problems in Engineering

where 120585119894 119894 = 1 2 3 and 120578 depend on 119909 119910 119905 and 119906 is a Liepoint symmetry of (1) if

pr(4)119883[(119906119905+ 120572 (119906

119899)119909+ 120573119906119909119909119905

)119909+ 120574119906119910119910] = 0 (3)

whenever (119906119905+ 120572(119906119899)

119909+ 120573119906119909119909119905

)119909+ 120574119906119910119910

= 0 Here pr(4)119883[20] denotes the fourth prolongation of119883 Expanding (3) andsplitting on the derivatives of 119906 we obtain an overdeterminedsystem of linear partial differential equations Solving thissystem one obtains the following four Lie point symmetries

1198831=

120597

120597119909

1198832=

120597

120597119910

1198833=

120597

120597119905

1198834= 119910 (1 minus 119899)

120597

120597119910+ 2119905 (1 minus 119899)

120597

120597119905+ 2119906

120597

120597119906

(4)

21 One-Dimensional Optimal System of Subalgebras Wenow calculate the optimal system of one-dimensional subal-gebras for (1) and use it to find the optimal system of group-invariant solutions for (1) We follow the method given in[20] Recall that the adjoint transformations are given by

Ad (exp (120598119883119894))119883119895= 119883119895minus 120598 [119883

119894 119883119895]

+1

21205982[119883119894[119883119894 119883119895]] minus sdot sdot sdot

(5)

where [119883119894 119883119895] is the commutator defined by

[119883119894 119883119895] = 119883

119894119883119895minus 119883119895119883119894 (6)

We present the commutator table of the Lie symmetriesand the adjoint representations of the symmetry group of(1) on its Lie algebra in Tables 1 and 2 respectively Thesetwo tables are then used to construct the optimal systemof one-dimensional subalgebras for (1) As a result aftersome calculations one can obtain an optimal system of one-dimensional subalgebras given by 119886119883

1+1198871198832+1198881198833 1198891198831+1198834

where 119886 119889 isin R 119887 119888 = 0 plusmn1

22 Symmetry Reductions and Exact Solutions of (1) In thissubsection we use the optimal system of one-dimensionalsubalgebras calculated above to obtain symmetry reductionsand exact solutions of the KP-MEW equation

Case 1 Consider the following 1198861198831+1198871198832+1198881198833 119886 isin R 119887 119888 =

plusmn1

The symmetry 1198861198831+1198871198832+1198881198833gives rise to the following

three invariants

119891 =119887119905 minus 119888119910

119887 119892 =

119887119909 + 119886119910

119887 120601 = 119906 (7)

Table 1 Commutator table of the Lie algebra of equation (1)

1198831

1198832

1198833

1198834

1198831

0 0 0 0

1198832

0 0 0 (1 minus 119899)1198832

1198833

0 0 0 (2 minus 2119899)1198833

1198834

0 minus(1 minus 119899)1198832

minus(2 minus 2119899)1198833

0

Table 2 Adjoint table of the Lie algebra of equation (1)

Ad 1198831

1198832

1198833

1198834

1198831

1198831

1198832

1198833

1198834

1198832

1198831

1198832

1198833

1198834minus 120576(1 minus 119899)119883

2

1198833

1198831

1198832

1198833

1198834minus 120576(2 minus 2119899)119883

3

1198834

1198831

119890120576(1minus119899)1198832

119890120576(2minus2119899)1198833

1198834

Now treating 120601 as the new dependent variable and 119891 and119892 as new independent variables the KP-MEW equation (1)transforms to

(2119886119888120574 + 1198872) 120601119891119892

+ 1198862120574120601119892119892

+ 120572119899 (119899 minus 1198872) 120601119899minus2

1206012

119892

+ 1198872120572119899120601119899minus1

120601119892119892

+ 1198872120573120601119891119892119892119892

+ 1198882120574120601119891119891

= 0

(8)

which is a nonlinear PDE in two independent variables Wenow use the Lie point symmetries of (8) and transform it toan ordinary differential equation (ODE) Equation (8) has thetwo translational symmetries namely

Γ1=

120597

120597119891 Γ

2=

120597

120597119892 (9)

The combination Γ1+Γ2of the two symmetries Γ

1and Γ2yields

the two invariants

119911 = 119891 minus 119892 119865 = 120601 (10)

which gives rise to a group-invariant solution 119865 = 119865(119911)Consequently using these invariants (8) is transformed intothe fourth-order nonlinear ODE

120574 (119886 + 119888)211986510158401015840+ 120572119899119887

2119865119899minus1

11986510158401015840minus 119887211986510158401015840

+ 1205721198991198872

(119899 minus 1) 119865119899minus2

11986510158402minus 12057311988721198651015840101584010158401015840

= 0

(11)

Integrating the above equation twice and taking the constantsof integration to be zero we obtain a second-order ODE

(120574 (119886 + 119888)2minus 1198882) 119865 + 120572119887

2119865119899minus 120573119887211986510158401015840= 0 (12)

Multiplying (12) by 1198651015840 integrating once and taking the

constant of integration to be zero we obtain the first-orderODE

1

2(120574 (119886 + 119888)

2minus 1198872) 1198652+

1205721198872

119899 + 1119865119899+1

minus1

2120573119887211986510158402

= 0 (13)

One can integrate the above equation by separating thevariables After integrating and reverting back to the originalvariables we obtain the following group-invariant solutions

Mathematical Problems in Engineering 3

of the KP-MEW equation (1) for arbitrary values of 119899 in thefollowing form

119906 (119909 119910 119905)

= ((119899 + 1)(1198872 minus 120574 (119886 + 119888)

2)

21205731198872)

1(119899minus1)

sech2(119899minus1) [119876]

(14)

where

119876 =radic120574 (119886 + 119888)

2+ 1198872

2radic2(radic2 (1 minus 119899)

119888radic120573119911 + (119899 minus 1) 119862)

119911 = 119905 minus 119909 minus(119886 + 119888)

119887119910

(15)

and 119862 is a constant of integration By taking 119899 = 2 120572 = 12120573 = 1 120574 = 1 119886 = 1 119887 = 1 119888 = 1 119905 = 0 and 119862 = 1 in (14) theprofile of the solution is given in Figure 1

Case 2 Consider the following 1198891198831+ 1198834

The symmetry 1198891198831+1198834gives rise to the three invariants

119891 =119905

1199102 119892 =

119889 ln119910 + 119899119909 minus 119909

119899 minus 1 120601 = 119906119910

2(119899minus1)

(16)

By treating 120601 as the new dependent variable and 119891 and 119892

as new independent variables the KP-MEW equation (1)transforms to

1205731198992120601119891119892119892119892

minus 2120573119899120601119891119892119892119892

+ 2119889120574120601119892119892

+ 2119899120574120601 + 41198912120574120601119891119891

minus 2119891120574120601119891minus 31198861120574120601119892+ 120601119891119892

minus 120572119899120601119899minus2

1206012

119892

+ 1205721198994120601119899minus2

1206012

119892+ 3120572119899

2120601119899minus2

1206012

119892+ 6119891119899

2120574120601119891

minus 4119891119899120574120601119891minus 3120572119899

3120601119899minus2

1206012

119892+ 120572119899120601

119899minus1120601119892119892

+ 1205721198993120601119899minus1

120601119892119892

minus 81198912119899120574120601119891119891

+ 411989121198992120574120601119891119891

+ 4119889119891120574120601119891119892

minus 21205721198992120601119899minus1

120601119892119892

+ 2120574120601 + 120573120601119891119892119892119892

+ 1198911198992120601119892minus 2119891119899120601

119892minus 4119889119891119899120574120601

119891119892minus 119889119891119899120574120601

119892= 0

(17)

Equation (17) has a single Lie point symmetry namely

Γ1=

120597

120597119892 (18)

and this symmetry yields the two invariants

119903 = 119891 119865 = 120601 (19)

which gives rise to a group-invariant solution 119865 = 119865(119903) andconsequently using these invariants (17) is then transformedto a second-order Cauchy-Euler ODE

(211989921199032minus 41198991199032+ 21199032) 11986510158401015840+ (31198992119903 minus 2119899119903 minus 119903) 119865

1015840

+ (1 + 119899) 119865 = 0

(20)

2

2

0

0

0

minus2

minus2

minus2

minus4

minus6

minus8

y

x

u

Figure 1 Profile of solution (14)

Now solving this equation and reverting back to the originalvariables we obtain the following solution of the KP-MEWequation (1)

119906 (119909 119910 119905) =1

1199102(119899minus1)(1198621119903minus(119899+1)2(119899minus1)

+ 1198622119903minus1(119899minus1)

) (21)

where 119903 = 1199051199102 and 1198621and 119862

2are constants of integration

3 (1198661015840119866)-Expansion Method

In this section we use the (1198661015840119866)-expansion method [9 30]to obtain a few exact solutions of the KP-MEW equation (1)for 119899 = 2 and 119899 = 3

Let us consider the solutions of (11) in the form

119865 (119911) =

119872

sum119894=0

A119894(1198661015840 (119911)

119866 (119911))

119894

(22)

where 119866(119911) satisfies

11986610158401015840+ 1205821198661015840+ 120583119866 = 0 (23)

and 120582 and 120583 are constantsThe homogeneous balancemethodbetween the highest order derivative and highest ordernonlinear term appearing in (11) determines the value of 119872andA

0 A

119872are constants to be determined

Consider 119899 = 2 Application of the balancing procedure tofourth-order ODE (11) yields119872 = 2 so the solution of (11) isof the form

119865 (119911) = A0+A1(1198661015840 (119911)

119866 (119911)) +A

2(1198661015840 (119911)

119866 (119911))

2

(24)

4 Mathematical Problems in Engineering

Substituting (23) and (24) into (11) leads to an overdeterminedsystem of algebraic equations Solving this system of algebraicequations with the aid of Maple we obtain

A0=

119887212057212058221198602+ 81198872120583120572119860

2minus 61205741198862 minus 12119886119888120574 minus 61205741198882 + 61198872

121205721198872

A1=

6120573120582

120572

A2=

6120573

120572

(25)

Now using the general solution of (23) in (24) we have thefollowing three types of travelling wave solutions of the KP-MEW equation (1)

When 1205822 minus 4120583 gt 0 we obtain the hyperbolic function

solution

119906 (119909 119910 119905)

= A0+A1(minus

120582

2+ 1205751

1198621sinh (120575

1119911) + 119862

2cosh (120575

1119911)

1198621cosh (120575

1119911) + 119862

2sinh (120575

1119911)

)

+A2(minus

120582

2+ 1205751

1198621sinh (120575

1119911) + 119862

2cosh (120575

1119911)

1198621cosh (120575

1119911) + 119862

2sinh (120575

1119911)

)

2

(26)

where 119911 = 119905 minus 119909 minus ((119886 + 119888)119887)119910 1205751= (12)radic1205822 minus 4120583 and 119862

1

and 1198622are arbitrary constants

The profile of the solution (26) is given in Figure 2When 1205822 minus 4120583 lt 0 we obtain the trigonometric function

solution

119906 (119909 119910 119905)

= A0+A1(minus

120582

2+ 1205752

minus1198621sin (1205752119911) + 119862

2cos (120575

2119911)

1198621cos (120575

2119911) + 119862

2sin (1205752119911)

)

+A2(minus

120582

2+ 1205752

minus1198621sin (1205752119911) + 119862

2cos (120575

2119911)

1198621cos (120575

2119911) + 119862

2sin (1205752119911)

)

2

(27)

where 119911 = 119905 minus 119909 minus ((119886 + 119888)119887)119910 1205752= (12)radic4120583 minus 1205822 and 119862

1

and 1198622are arbitrary constants

The profile of the solution (27) is given in Figure 3When 1205822 minus 4120583 = 0 we obtain the rational function

solution

119906 (119909 119910 119905) = A0+A1(minus

120582

2+

1198622

1198621+ 1198622119911)

+A2(minus

120582

2+

1198622

1198621+ 1198622119911)

2

(28)

where 119911 = 119905 minus 119909 minus ((119886 + 119888)119887)119910 and 1198621and 119862

2are arbitrary

constantsThe profile of the solution (28) is given in Figure 4

2

2

0

0

010

20

30

40

minus2

minus2

y

x

u

Figure 2 Profile of solution (26)

2

2

0

0

500

1000

1500

0

minus2

minus2

y

x

u

Figure 3 Profile of solution (27)

2

2

0

0

0

10

20

minus2

minus2

y

x

u

Figure 4 Profile of solution (28)

Mathematical Problems in Engineering 5

Consider 119899 = 3 Again the application of the balancingprocedure to fourth-order ODE yields119872 = 1 so the solutionof (11) is of the form

119865 (119911) = A0+A1(1198661015840 (119911)

119866 (119911)) (29)

Solving this system of algebraic equations with the aid ofMaple we obtain

120573 =120572

2A2

1

A0=

120582radicminus1198862120574 minus 2119886119888120574 + 1198872 minus 1198882120574

radic12057211988721205822 minus 41205721198872120583

A1=

2radicminus1198862120574 minus 2119886119888120574 + 1198872 minus 1198882120574

radic12057211988721205822 minus 41205721198872120583

(30)

Now using the general solution of (23) in (29) we have thefollowing two types of travelling wave solutions of the KP-MEW equation (1)

When 1205822 minus 4120583 gt 0 we obtain the hyperbolic function

solution

119906 (119909 119910 119905)

= A0+A1(minus

120582

2+ 1205751

1198621sinh (120575

1119911) + 119862

2cosh (120575

1119911)

1198621cosh (120575

1119911) + 119862

2sinh (120575

1119911)

)

(31)

where 119911 = 119905 minus 119909 minus ((119886 + 119888)119887)119910 1205751= (12)radic1205822 minus 4120583 and 119862

1

and 1198622are arbitrary constants

When 1205822 minus 4120583 lt 0 we obtain the trigonometric functionsolution

119906 (119909 119910 119905)

= A0+A1(minus

120582

2+ 1205752

minus1198621sin (1205752119911) + 119862

2cos (120575

2119911)

1198621cos (120575

2119911) + 119862

2sin (1205752119911)

)

(32)

where 119911 = 119905 minus 119909 minus ((119886 + 119888)119887)119910 1205752= (12)radic4120583 minus 1205822 and 119862

1

and 1198622are arbitrary constants

4 Conservation Laws of (1)

In this section we construct conservation laws for (1) Themultiplier method [29 30] will be used

The zeroth-order multiplierΛ(119905 119909 119910 119906) for (1) is given by

Λ = minus1199103

61205741198911015840

1(119905) + 119909119910119891

1(119905) minus

1199102

21205741198911015840

2(119905)

+ 1199091198912(119905) + 119910119891

3(119905) + 119891

4(119905)

(33)

where1198911(119905)1198912(119905)1198913(119905) and119891

4(119905) are arbitrary functions of 119905

Corresponding to the above multiplier we have the followingconserved vectors of (1)

119879119905

1=

1

24120574minus12120574119910119891

1(119905) 119906 minus 6120573120574119910119891

1(119905) 119906119909119909

+ 61205731205741199091199101198911(119905) 119906119909119909119909

+ 121205741199091199101198911(119905) 119906119909

minus12057311991031198911015840

1(119905) 119906119909119909119909

minus 211991031198911015840

1(119905) 119906119909

119879119909

1= minus

119910

241205741199064120572119899119910

21198911015840

1(119905) 119906119909119906119899minus 24120572120574119899119909119891

1(119905) 119906119909119906119899

minus 120573119910211989110158401015840

1(119905) 119906119909119909119906 + 3120573119910

21198911015840

1(119905) 119906119905119909119909

119906

+ 211991021198911015840

1(119905) 119906119905119906 minus 12120573120574119891

1015840

1(119905) 119906119909119906

+ 61205731205741199091198911015840

1(119905) 119906119909119909119906 + 12120573120574119891

1(119905) 119906119905119909119906

minus 181205731205741199091198911(119905) 119906119905119909119909

119906 minus 121205741199091198911(119905) 119906119905119906

+ 241205721205741198911(119905) 119906119899+1

minus 2119910211989110158401015840

1(119905) 1199062

+ 121205741199091198911015840

1(119905) 1199062

119879119910

1=

1

6311991021198911015840

1(119905) 119906 minus 6120574119909119891

1(119905) 119906

+ 61205741199091199101198911(119905) 119906119910minus 11991031198911015840

1(119905) 119906119910

119879119905

2=

1

8120574minus4120574119891

2(119905) 119906 minus 2120573120574119891

2(119905) 119906119909119909

+ 21205731205741199091198912(119905) 119906119909119909119909

+ 41205741199091198912(119905) 119906119909minus 12057311991021198911015840

2(119905) 119906119909119909119909

minus 211991021198911015840

2(119905) 119906119909

119879119909

2= minus

1

81205741199064120572119899119910

21198911015840

2(119905) 119906119909119906119899minus 8120572120574119899119909119891

2(119905) 119906119909119906119899

minus 120573119910211989110158401015840

2119906119909119909119906 + 3120573119910

21198911015840

2119906119905119909119909

119906

+ 211991021198911015840

2(119905) 119906119905119906 minus 4120573120574119891

1015840

2(119905) 119906119909119906

+ 21205731205741199091198911015840

2(119905) 119906119909119909119906 + 4120573120574119891

2(119905) 119906119905119909119906

minus 61205731205741199091198912(119905) 119906119905119909119909

119906 minus 41205741199091198912(119905) 119906119905119906

+ 81205721205741198912(119905) 119906119899+1

minus 2119910211989110158401015840

2(119905) 1199062

+ 41205741199091198911015840

2(119905) 1199062

119879119910

2=

1

221199101198911015840

2(119905) 119906 + 2120574119909119891

2(119905) 119906119910minus 11991021198911015840

2(119905) 119906119910

119879119905

3=

1

41205731199101198913(119905) 119906119909119909119909

+ 21199101198913(119905) 119906119909

119879119909

3= minus

119910

4119906minus4120572119899119891

3(119905) 119906119909119906119899+ 1205731198911015840

3(119905) 119906119909119909119906 minus 3120573119891

3(119905) 119906119905119909119909

119906

minus 21198913(119905) 119906119905119906 + 2119891

1015840

3(119905) 1199062

119879119910

3= 120574119910119891

3(119905) 119906119910minus 1205741198913(119905) 119906

6 Mathematical Problems in Engineering

119879119905

4=

1

41205731198914(119905) 119906119909119909119909

+ 21198914(119905) 119906119909

119879119909

4= minus

1

4119906minus4120572119899119891

4(119905) 119906119909119906119899+ 1205731198911015840

4(119905) 119906119909119909119906 minus 3120573119891

4(119905) 119906119905119909119909

119906

minus 21198914(119905) 119906119905119906 + 2119891

1015840

4(119905) 1199062

119879119910

4= 1205741198914(119905) 119906119910

(34)

Remark The presence of the arbitrary functions in themultiplier leads to a family of infinitely many conservationlaws for (1)

5 Concluding Remarks

In this paper we obtained the solutions of a gener-alized two-dimensional nonlinear Kadomtsev-Petviashvili-modified equal width equation by employing the Lie groupanalysis the optimal systems of one-dimensional subal-gebras and the (119866

1015840119866)-expansion method The solutionsobtained are solitary waves and nontopological solitons Theconservation laws for the underlying equation were alsoderived by using the multiplier method

Conflict of Interests

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

References

[1] M J Ablowitz and P A Clarkson Soliton Nonlinear EvolutionEquations and Inverse Scattering Cambridge University PressCambridge UK 1991

[2] M Wang Y Zhou and Z Li ldquoApplication of a homogeneousbalance method to exact solutions of nonlinear equations inmathematical physicsrdquo Physics Letters A vol 216 no 1ndash5 pp67ndash75 1996

[3] C H Gu Soliton Theory and Its Application Zhejiang Scienceand Technology Press Zhejiang China 1990

[4] Y Chen and Z Yan ldquoThe Weierstrass elliptic function expan-sion method and its applications in nonlinear wave equationsrdquoChaos Solitons and Fractals vol 29 no 4 pp 948ndash964 2006

[5] V B Matveev and M A Salle Darboux Transformation andSoliton Springer Berlin Germany 1991

[6] J Hu ldquoExplicit solutions to three nonlinear physical modelsrdquoPhysics Letters A vol 287 no 1-2 pp 81ndash89 2001

[7] J Hu and H Zhang ldquoA new method for finding exact travelingwave solutions to nonlinear partial differential equationsrdquoPhysics Letters A vol 286 no 2-3 pp 175ndash179 2001

[8] R Hirota The Direct Method in Soliton Theory vol 155 ofCambridge Tracts in Mathematics Cambridge University PressCambridge UK 2004

[9] M Wang X Li and J Zhang ldquoThe (1198661015840

119866)-expansion methodand travelling wave solutions of nonlinear evolution equationsin mathematical physicsrdquo Physics Letters A vol 372 no 4 pp417ndash423 2008

[10] D Lu ldquoJacobi elliptic function solutions for two variant Boussi-nesq equationsrdquo Chaos Solitons amp Fractals vol 24 no 5 pp1373ndash1385 2005

[11] Z Yan ldquoAbundant families of Jacobi elliptic function solutionsof the (2 + 1)-dimensional integrable Davey-Stewartson-typeequation via a new methodrdquo Chaos Solitons and Fractals vol18 no 2 pp 299ndash309 2003

[12] S-Y Lou and J Lu ldquoSpecial solutions from the variableseparation approach the Davey-Stewartson equationrdquo Journalof Physics A Mathematical and General vol 29 no 14 pp4209ndash4215 1996

[13] A-M Wazwaz ldquoThe tanh and the sine-cosine methods forcompact and noncompact solutions of the nonlinear Klein-Gordon equationrdquo Applied Mathematics and Computation vol167 no 2 pp 1179ndash1195 2005

[14] Z Yan ldquoThe new tri-function method to multiple exact solu-tions of nonlinear wave equationsrdquo Physica Scripta vol 78 no3 Article ID 035001 5 pages 2008

[15] Z Yan ldquoPeriodic solitary and rational wave solutions of the3D extended quantum Zakharov-Kuznetsov equation in densequantum plasmasrdquo Physics Letters Section A General Atomicand Solid State Physics vol 373 no 29 pp 2432ndash2437 2009

[16] M Wang and X Li ldquoExtended 119865-expansion method and peri-odic wave solutions for the generalized Zakharov equationsrdquoPhysics Letters A vol 343 no 1ndash3 pp 48ndash54 2005

[17] S Zhang ldquoApplication of Exp-function method to high-dimensional nonlinear evolution equationrdquo Chaos Solitons andFractals vol 38 no 1 pp 270ndash276 2008

[18] W-X Ma T Huang and Y Zhang ldquoA multiple exp-functionmethod for nonlinear differential equations and its applicationrdquoPhysica Scripta vol 82 no 6 Article ID 065003 2010

[19] G W Bluman and S Kumei Symmetries and DifferentialEquations AppliedMathematical Sciences Springer NewYorkNY USA 1989

[20] P J Olver Applications of Lie Groups to Differential Equationsvol 107 of Graduate Texts in Mathematics Springer New YorkNY USA 2nd edition 1993

[21] L V Ovsiannikov Group Analysis of Differential EquationsAcademic Press New York NY USA 1982

[22] N H Ibragimov CRC Handbook of Lie Group Analysis ofDifferential Equations vol 1 CRC Press Boca Raton Fla USA1994

[23] N H Ibragimov CRC Handbook of Lie Group Analysis ofDifferential Equations vol 2 CRC Press Boca Raton Fla USA1995

[24] N H Ibragimov CRC Handbook of Lie Group Analysis ofDifferential Equations vol 3 CRC Press Boca Raton Fla USA1996

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

[26] A-M Wazwaz ldquoThe tanh method and the sine-cosine methodfor solving the KP-MEW equationrdquo International Journal ofComputer Mathematics vol 82 no 2 pp 235ndash246 2005

[27] A Saha ldquoBifurcation of travelling wave solutions for the gener-alized KP-NEW equationsrdquo Communications in Nonlinear Sci-ence and Numerical Simulation vol 17 no 9 pp 3539ndash35512012

[28] MWei S Tang H Fu and G Chen ldquoSingle peak solitary wavesolutions for the generalized KP-MEW (2 2) equation under

Mathematical Problems in Engineering 7

boundary conditionrdquo Applied Mathematics and Computationvol 219 no 17 pp 8979ndash8990 2013

[29] S C Anco and G Bluman ldquoDirect construction method forconservation laws of partial differential equations Part I exam-ples of conservation law classificationsrdquo European Journal ofApplied Mathematics vol 13 no 5 pp 545ndash566 2002

[30] K R Adem and C M Khalique ldquoConservation laws andtraveling wave solutions of a generalized nonlinear ZK-BBMequationrdquo Abstract and Applied Analysis vol 2014 Article ID139513 5 pages 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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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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

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

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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 Mathematical Problems in Engineering

where 120585119894 119894 = 1 2 3 and 120578 depend on 119909 119910 119905 and 119906 is a Liepoint symmetry of (1) if

pr(4)119883[(119906119905+ 120572 (119906

119899)119909+ 120573119906119909119909119905

)119909+ 120574119906119910119910] = 0 (3)

whenever (119906119905+ 120572(119906119899)

119909+ 120573119906119909119909119905

)119909+ 120574119906119910119910

= 0 Here pr(4)119883[20] denotes the fourth prolongation of119883 Expanding (3) andsplitting on the derivatives of 119906 we obtain an overdeterminedsystem of linear partial differential equations Solving thissystem one obtains the following four Lie point symmetries

1198831=

120597

120597119909

1198832=

120597

120597119910

1198833=

120597

120597119905

1198834= 119910 (1 minus 119899)

120597

120597119910+ 2119905 (1 minus 119899)

120597

120597119905+ 2119906

120597

120597119906

(4)

21 One-Dimensional Optimal System of Subalgebras Wenow calculate the optimal system of one-dimensional subal-gebras for (1) and use it to find the optimal system of group-invariant solutions for (1) We follow the method given in[20] Recall that the adjoint transformations are given by

Ad (exp (120598119883119894))119883119895= 119883119895minus 120598 [119883

119894 119883119895]

+1

21205982[119883119894[119883119894 119883119895]] minus sdot sdot sdot

(5)

where [119883119894 119883119895] is the commutator defined by

[119883119894 119883119895] = 119883

119894119883119895minus 119883119895119883119894 (6)

We present the commutator table of the Lie symmetriesand the adjoint representations of the symmetry group of(1) on its Lie algebra in Tables 1 and 2 respectively Thesetwo tables are then used to construct the optimal systemof one-dimensional subalgebras for (1) As a result aftersome calculations one can obtain an optimal system of one-dimensional subalgebras given by 119886119883

1+1198871198832+1198881198833 1198891198831+1198834

where 119886 119889 isin R 119887 119888 = 0 plusmn1

22 Symmetry Reductions and Exact Solutions of (1) In thissubsection we use the optimal system of one-dimensionalsubalgebras calculated above to obtain symmetry reductionsand exact solutions of the KP-MEW equation

Case 1 Consider the following 1198861198831+1198871198832+1198881198833 119886 isin R 119887 119888 =

plusmn1

The symmetry 1198861198831+1198871198832+1198881198833gives rise to the following

three invariants

119891 =119887119905 minus 119888119910

119887 119892 =

119887119909 + 119886119910

119887 120601 = 119906 (7)

Table 1 Commutator table of the Lie algebra of equation (1)

1198831

1198832

1198833

1198834

1198831

0 0 0 0

1198832

0 0 0 (1 minus 119899)1198832

1198833

0 0 0 (2 minus 2119899)1198833

1198834

0 minus(1 minus 119899)1198832

minus(2 minus 2119899)1198833

0

Table 2 Adjoint table of the Lie algebra of equation (1)

Ad 1198831

1198832

1198833

1198834

1198831

1198831

1198832

1198833

1198834

1198832

1198831

1198832

1198833

1198834minus 120576(1 minus 119899)119883

2

1198833

1198831

1198832

1198833

1198834minus 120576(2 minus 2119899)119883

3

1198834

1198831

119890120576(1minus119899)1198832

119890120576(2minus2119899)1198833

1198834

Now treating 120601 as the new dependent variable and 119891 and119892 as new independent variables the KP-MEW equation (1)transforms to

(2119886119888120574 + 1198872) 120601119891119892

+ 1198862120574120601119892119892

+ 120572119899 (119899 minus 1198872) 120601119899minus2

1206012

119892

+ 1198872120572119899120601119899minus1

120601119892119892

+ 1198872120573120601119891119892119892119892

+ 1198882120574120601119891119891

= 0

(8)

which is a nonlinear PDE in two independent variables Wenow use the Lie point symmetries of (8) and transform it toan ordinary differential equation (ODE) Equation (8) has thetwo translational symmetries namely

Γ1=

120597

120597119891 Γ

2=

120597

120597119892 (9)

The combination Γ1+Γ2of the two symmetries Γ

1and Γ2yields

the two invariants

119911 = 119891 minus 119892 119865 = 120601 (10)

which gives rise to a group-invariant solution 119865 = 119865(119911)Consequently using these invariants (8) is transformed intothe fourth-order nonlinear ODE

120574 (119886 + 119888)211986510158401015840+ 120572119899119887

2119865119899minus1

11986510158401015840minus 119887211986510158401015840

+ 1205721198991198872

(119899 minus 1) 119865119899minus2

11986510158402minus 12057311988721198651015840101584010158401015840

= 0

(11)

Integrating the above equation twice and taking the constantsof integration to be zero we obtain a second-order ODE

(120574 (119886 + 119888)2minus 1198882) 119865 + 120572119887

2119865119899minus 120573119887211986510158401015840= 0 (12)

Multiplying (12) by 1198651015840 integrating once and taking the

constant of integration to be zero we obtain the first-orderODE

1

2(120574 (119886 + 119888)

2minus 1198872) 1198652+

1205721198872

119899 + 1119865119899+1

minus1

2120573119887211986510158402

= 0 (13)

One can integrate the above equation by separating thevariables After integrating and reverting back to the originalvariables we obtain the following group-invariant solutions

Mathematical Problems in Engineering 3

of the KP-MEW equation (1) for arbitrary values of 119899 in thefollowing form

119906 (119909 119910 119905)

= ((119899 + 1)(1198872 minus 120574 (119886 + 119888)

2)

21205731198872)

1(119899minus1)

sech2(119899minus1) [119876]

(14)

where

119876 =radic120574 (119886 + 119888)

2+ 1198872

2radic2(radic2 (1 minus 119899)

119888radic120573119911 + (119899 minus 1) 119862)

119911 = 119905 minus 119909 minus(119886 + 119888)

119887119910

(15)

and 119862 is a constant of integration By taking 119899 = 2 120572 = 12120573 = 1 120574 = 1 119886 = 1 119887 = 1 119888 = 1 119905 = 0 and 119862 = 1 in (14) theprofile of the solution is given in Figure 1

Case 2 Consider the following 1198891198831+ 1198834

The symmetry 1198891198831+1198834gives rise to the three invariants

119891 =119905

1199102 119892 =

119889 ln119910 + 119899119909 minus 119909

119899 minus 1 120601 = 119906119910

2(119899minus1)

(16)

By treating 120601 as the new dependent variable and 119891 and 119892

as new independent variables the KP-MEW equation (1)transforms to

1205731198992120601119891119892119892119892

minus 2120573119899120601119891119892119892119892

+ 2119889120574120601119892119892

+ 2119899120574120601 + 41198912120574120601119891119891

minus 2119891120574120601119891minus 31198861120574120601119892+ 120601119891119892

minus 120572119899120601119899minus2

1206012

119892

+ 1205721198994120601119899minus2

1206012

119892+ 3120572119899

2120601119899minus2

1206012

119892+ 6119891119899

2120574120601119891

minus 4119891119899120574120601119891minus 3120572119899

3120601119899minus2

1206012

119892+ 120572119899120601

119899minus1120601119892119892

+ 1205721198993120601119899minus1

120601119892119892

minus 81198912119899120574120601119891119891

+ 411989121198992120574120601119891119891

+ 4119889119891120574120601119891119892

minus 21205721198992120601119899minus1

120601119892119892

+ 2120574120601 + 120573120601119891119892119892119892

+ 1198911198992120601119892minus 2119891119899120601

119892minus 4119889119891119899120574120601

119891119892minus 119889119891119899120574120601

119892= 0

(17)

Equation (17) has a single Lie point symmetry namely

Γ1=

120597

120597119892 (18)

and this symmetry yields the two invariants

119903 = 119891 119865 = 120601 (19)

which gives rise to a group-invariant solution 119865 = 119865(119903) andconsequently using these invariants (17) is then transformedto a second-order Cauchy-Euler ODE

(211989921199032minus 41198991199032+ 21199032) 11986510158401015840+ (31198992119903 minus 2119899119903 minus 119903) 119865

1015840

+ (1 + 119899) 119865 = 0

(20)

2

2

0

0

0

minus2

minus2

minus2

minus4

minus6

minus8

y

x

u

Figure 1 Profile of solution (14)

Now solving this equation and reverting back to the originalvariables we obtain the following solution of the KP-MEWequation (1)

119906 (119909 119910 119905) =1

1199102(119899minus1)(1198621119903minus(119899+1)2(119899minus1)

+ 1198622119903minus1(119899minus1)

) (21)

where 119903 = 1199051199102 and 1198621and 119862

2are constants of integration

3 (1198661015840119866)-Expansion Method

In this section we use the (1198661015840119866)-expansion method [9 30]to obtain a few exact solutions of the KP-MEW equation (1)for 119899 = 2 and 119899 = 3

Let us consider the solutions of (11) in the form

119865 (119911) =

119872

sum119894=0

A119894(1198661015840 (119911)

119866 (119911))

119894

(22)

where 119866(119911) satisfies

11986610158401015840+ 1205821198661015840+ 120583119866 = 0 (23)

and 120582 and 120583 are constantsThe homogeneous balancemethodbetween the highest order derivative and highest ordernonlinear term appearing in (11) determines the value of 119872andA

0 A

119872are constants to be determined

Consider 119899 = 2 Application of the balancing procedure tofourth-order ODE (11) yields119872 = 2 so the solution of (11) isof the form

119865 (119911) = A0+A1(1198661015840 (119911)

119866 (119911)) +A

2(1198661015840 (119911)

119866 (119911))

2

(24)

4 Mathematical Problems in Engineering

Substituting (23) and (24) into (11) leads to an overdeterminedsystem of algebraic equations Solving this system of algebraicequations with the aid of Maple we obtain

A0=

119887212057212058221198602+ 81198872120583120572119860

2minus 61205741198862 minus 12119886119888120574 minus 61205741198882 + 61198872

121205721198872

A1=

6120573120582

120572

A2=

6120573

120572

(25)

Now using the general solution of (23) in (24) we have thefollowing three types of travelling wave solutions of the KP-MEW equation (1)

When 1205822 minus 4120583 gt 0 we obtain the hyperbolic function

solution

119906 (119909 119910 119905)

= A0+A1(minus

120582

2+ 1205751

1198621sinh (120575

1119911) + 119862

2cosh (120575

1119911)

1198621cosh (120575

1119911) + 119862

2sinh (120575

1119911)

)

+A2(minus

120582

2+ 1205751

1198621sinh (120575

1119911) + 119862

2cosh (120575

1119911)

1198621cosh (120575

1119911) + 119862

2sinh (120575

1119911)

)

2

(26)

where 119911 = 119905 minus 119909 minus ((119886 + 119888)119887)119910 1205751= (12)radic1205822 minus 4120583 and 119862

1

and 1198622are arbitrary constants

The profile of the solution (26) is given in Figure 2When 1205822 minus 4120583 lt 0 we obtain the trigonometric function

solution

119906 (119909 119910 119905)

= A0+A1(minus

120582

2+ 1205752

minus1198621sin (1205752119911) + 119862

2cos (120575

2119911)

1198621cos (120575

2119911) + 119862

2sin (1205752119911)

)

+A2(minus

120582

2+ 1205752

minus1198621sin (1205752119911) + 119862

2cos (120575

2119911)

1198621cos (120575

2119911) + 119862

2sin (1205752119911)

)

2

(27)

where 119911 = 119905 minus 119909 minus ((119886 + 119888)119887)119910 1205752= (12)radic4120583 minus 1205822 and 119862

1

and 1198622are arbitrary constants

The profile of the solution (27) is given in Figure 3When 1205822 minus 4120583 = 0 we obtain the rational function

solution

119906 (119909 119910 119905) = A0+A1(minus

120582

2+

1198622

1198621+ 1198622119911)

+A2(minus

120582

2+

1198622

1198621+ 1198622119911)

2

(28)

where 119911 = 119905 minus 119909 minus ((119886 + 119888)119887)119910 and 1198621and 119862

2are arbitrary

constantsThe profile of the solution (28) is given in Figure 4

2

2

0

0

010

20

30

40

minus2

minus2

y

x

u

Figure 2 Profile of solution (26)

2

2

0

0

500

1000

1500

0

minus2

minus2

y

x

u

Figure 3 Profile of solution (27)

2

2

0

0

0

10

20

minus2

minus2

y

x

u

Figure 4 Profile of solution (28)

Mathematical Problems in Engineering 5

Consider 119899 = 3 Again the application of the balancingprocedure to fourth-order ODE yields119872 = 1 so the solutionof (11) is of the form

119865 (119911) = A0+A1(1198661015840 (119911)

119866 (119911)) (29)

Solving this system of algebraic equations with the aid ofMaple we obtain

120573 =120572

2A2

1

A0=

120582radicminus1198862120574 minus 2119886119888120574 + 1198872 minus 1198882120574

radic12057211988721205822 minus 41205721198872120583

A1=

2radicminus1198862120574 minus 2119886119888120574 + 1198872 minus 1198882120574

radic12057211988721205822 minus 41205721198872120583

(30)

Now using the general solution of (23) in (29) we have thefollowing two types of travelling wave solutions of the KP-MEW equation (1)

When 1205822 minus 4120583 gt 0 we obtain the hyperbolic function

solution

119906 (119909 119910 119905)

= A0+A1(minus

120582

2+ 1205751

1198621sinh (120575

1119911) + 119862

2cosh (120575

1119911)

1198621cosh (120575

1119911) + 119862

2sinh (120575

1119911)

)

(31)

where 119911 = 119905 minus 119909 minus ((119886 + 119888)119887)119910 1205751= (12)radic1205822 minus 4120583 and 119862

1

and 1198622are arbitrary constants

When 1205822 minus 4120583 lt 0 we obtain the trigonometric functionsolution

119906 (119909 119910 119905)

= A0+A1(minus

120582

2+ 1205752

minus1198621sin (1205752119911) + 119862

2cos (120575

2119911)

1198621cos (120575

2119911) + 119862

2sin (1205752119911)

)

(32)

where 119911 = 119905 minus 119909 minus ((119886 + 119888)119887)119910 1205752= (12)radic4120583 minus 1205822 and 119862

1

and 1198622are arbitrary constants

4 Conservation Laws of (1)

In this section we construct conservation laws for (1) Themultiplier method [29 30] will be used

The zeroth-order multiplierΛ(119905 119909 119910 119906) for (1) is given by

Λ = minus1199103

61205741198911015840

1(119905) + 119909119910119891

1(119905) minus

1199102

21205741198911015840

2(119905)

+ 1199091198912(119905) + 119910119891

3(119905) + 119891

4(119905)

(33)

where1198911(119905)1198912(119905)1198913(119905) and119891

4(119905) are arbitrary functions of 119905

Corresponding to the above multiplier we have the followingconserved vectors of (1)

119879119905

1=

1

24120574minus12120574119910119891

1(119905) 119906 minus 6120573120574119910119891

1(119905) 119906119909119909

+ 61205731205741199091199101198911(119905) 119906119909119909119909

+ 121205741199091199101198911(119905) 119906119909

minus12057311991031198911015840

1(119905) 119906119909119909119909

minus 211991031198911015840

1(119905) 119906119909

119879119909

1= minus

119910

241205741199064120572119899119910

21198911015840

1(119905) 119906119909119906119899minus 24120572120574119899119909119891

1(119905) 119906119909119906119899

minus 120573119910211989110158401015840

1(119905) 119906119909119909119906 + 3120573119910

21198911015840

1(119905) 119906119905119909119909

119906

+ 211991021198911015840

1(119905) 119906119905119906 minus 12120573120574119891

1015840

1(119905) 119906119909119906

+ 61205731205741199091198911015840

1(119905) 119906119909119909119906 + 12120573120574119891

1(119905) 119906119905119909119906

minus 181205731205741199091198911(119905) 119906119905119909119909

119906 minus 121205741199091198911(119905) 119906119905119906

+ 241205721205741198911(119905) 119906119899+1

minus 2119910211989110158401015840

1(119905) 1199062

+ 121205741199091198911015840

1(119905) 1199062

119879119910

1=

1

6311991021198911015840

1(119905) 119906 minus 6120574119909119891

1(119905) 119906

+ 61205741199091199101198911(119905) 119906119910minus 11991031198911015840

1(119905) 119906119910

119879119905

2=

1

8120574minus4120574119891

2(119905) 119906 minus 2120573120574119891

2(119905) 119906119909119909

+ 21205731205741199091198912(119905) 119906119909119909119909

+ 41205741199091198912(119905) 119906119909minus 12057311991021198911015840

2(119905) 119906119909119909119909

minus 211991021198911015840

2(119905) 119906119909

119879119909

2= minus

1

81205741199064120572119899119910

21198911015840

2(119905) 119906119909119906119899minus 8120572120574119899119909119891

2(119905) 119906119909119906119899

minus 120573119910211989110158401015840

2119906119909119909119906 + 3120573119910

21198911015840

2119906119905119909119909

119906

+ 211991021198911015840

2(119905) 119906119905119906 minus 4120573120574119891

1015840

2(119905) 119906119909119906

+ 21205731205741199091198911015840

2(119905) 119906119909119909119906 + 4120573120574119891

2(119905) 119906119905119909119906

minus 61205731205741199091198912(119905) 119906119905119909119909

119906 minus 41205741199091198912(119905) 119906119905119906

+ 81205721205741198912(119905) 119906119899+1

minus 2119910211989110158401015840

2(119905) 1199062

+ 41205741199091198911015840

2(119905) 1199062

119879119910

2=

1

221199101198911015840

2(119905) 119906 + 2120574119909119891

2(119905) 119906119910minus 11991021198911015840

2(119905) 119906119910

119879119905

3=

1

41205731199101198913(119905) 119906119909119909119909

+ 21199101198913(119905) 119906119909

119879119909

3= minus

119910

4119906minus4120572119899119891

3(119905) 119906119909119906119899+ 1205731198911015840

3(119905) 119906119909119909119906 minus 3120573119891

3(119905) 119906119905119909119909

119906

minus 21198913(119905) 119906119905119906 + 2119891

1015840

3(119905) 1199062

119879119910

3= 120574119910119891

3(119905) 119906119910minus 1205741198913(119905) 119906

6 Mathematical Problems in Engineering

119879119905

4=

1

41205731198914(119905) 119906119909119909119909

+ 21198914(119905) 119906119909

119879119909

4= minus

1

4119906minus4120572119899119891

4(119905) 119906119909119906119899+ 1205731198911015840

4(119905) 119906119909119909119906 minus 3120573119891

4(119905) 119906119905119909119909

119906

minus 21198914(119905) 119906119905119906 + 2119891

1015840

4(119905) 1199062

119879119910

4= 1205741198914(119905) 119906119910

(34)

Remark The presence of the arbitrary functions in themultiplier leads to a family of infinitely many conservationlaws for (1)

5 Concluding Remarks

In this paper we obtained the solutions of a gener-alized two-dimensional nonlinear Kadomtsev-Petviashvili-modified equal width equation by employing the Lie groupanalysis the optimal systems of one-dimensional subal-gebras and the (119866

1015840119866)-expansion method The solutionsobtained are solitary waves and nontopological solitons Theconservation laws for the underlying equation were alsoderived by using the multiplier method

Conflict of Interests

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

References

[1] M J Ablowitz and P A Clarkson Soliton Nonlinear EvolutionEquations and Inverse Scattering Cambridge University PressCambridge UK 1991

[2] M Wang Y Zhou and Z Li ldquoApplication of a homogeneousbalance method to exact solutions of nonlinear equations inmathematical physicsrdquo Physics Letters A vol 216 no 1ndash5 pp67ndash75 1996

[3] C H Gu Soliton Theory and Its Application Zhejiang Scienceand Technology Press Zhejiang China 1990

[4] Y Chen and Z Yan ldquoThe Weierstrass elliptic function expan-sion method and its applications in nonlinear wave equationsrdquoChaos Solitons and Fractals vol 29 no 4 pp 948ndash964 2006

[5] V B Matveev and M A Salle Darboux Transformation andSoliton Springer Berlin Germany 1991

[6] J Hu ldquoExplicit solutions to three nonlinear physical modelsrdquoPhysics Letters A vol 287 no 1-2 pp 81ndash89 2001

[7] J Hu and H Zhang ldquoA new method for finding exact travelingwave solutions to nonlinear partial differential equationsrdquoPhysics Letters A vol 286 no 2-3 pp 175ndash179 2001

[8] R Hirota The Direct Method in Soliton Theory vol 155 ofCambridge Tracts in Mathematics Cambridge University PressCambridge UK 2004

[9] M Wang X Li and J Zhang ldquoThe (1198661015840

119866)-expansion methodand travelling wave solutions of nonlinear evolution equationsin mathematical physicsrdquo Physics Letters A vol 372 no 4 pp417ndash423 2008

[10] D Lu ldquoJacobi elliptic function solutions for two variant Boussi-nesq equationsrdquo Chaos Solitons amp Fractals vol 24 no 5 pp1373ndash1385 2005

[11] Z Yan ldquoAbundant families of Jacobi elliptic function solutionsof the (2 + 1)-dimensional integrable Davey-Stewartson-typeequation via a new methodrdquo Chaos Solitons and Fractals vol18 no 2 pp 299ndash309 2003

[12] S-Y Lou and J Lu ldquoSpecial solutions from the variableseparation approach the Davey-Stewartson equationrdquo Journalof Physics A Mathematical and General vol 29 no 14 pp4209ndash4215 1996

[13] A-M Wazwaz ldquoThe tanh and the sine-cosine methods forcompact and noncompact solutions of the nonlinear Klein-Gordon equationrdquo Applied Mathematics and Computation vol167 no 2 pp 1179ndash1195 2005

[14] Z Yan ldquoThe new tri-function method to multiple exact solu-tions of nonlinear wave equationsrdquo Physica Scripta vol 78 no3 Article ID 035001 5 pages 2008

[15] Z Yan ldquoPeriodic solitary and rational wave solutions of the3D extended quantum Zakharov-Kuznetsov equation in densequantum plasmasrdquo Physics Letters Section A General Atomicand Solid State Physics vol 373 no 29 pp 2432ndash2437 2009

[16] M Wang and X Li ldquoExtended 119865-expansion method and peri-odic wave solutions for the generalized Zakharov equationsrdquoPhysics Letters A vol 343 no 1ndash3 pp 48ndash54 2005

[17] S Zhang ldquoApplication of Exp-function method to high-dimensional nonlinear evolution equationrdquo Chaos Solitons andFractals vol 38 no 1 pp 270ndash276 2008

[18] W-X Ma T Huang and Y Zhang ldquoA multiple exp-functionmethod for nonlinear differential equations and its applicationrdquoPhysica Scripta vol 82 no 6 Article ID 065003 2010

[19] G W Bluman and S Kumei Symmetries and DifferentialEquations AppliedMathematical Sciences Springer NewYorkNY USA 1989

[20] P J Olver Applications of Lie Groups to Differential Equationsvol 107 of Graduate Texts in Mathematics Springer New YorkNY USA 2nd edition 1993

[21] L V Ovsiannikov Group Analysis of Differential EquationsAcademic Press New York NY USA 1982

[22] N H Ibragimov CRC Handbook of Lie Group Analysis ofDifferential Equations vol 1 CRC Press Boca Raton Fla USA1994

[23] N H Ibragimov CRC Handbook of Lie Group Analysis ofDifferential Equations vol 2 CRC Press Boca Raton Fla USA1995

[24] N H Ibragimov CRC Handbook of Lie Group Analysis ofDifferential Equations vol 3 CRC Press Boca Raton Fla USA1996

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

[26] A-M Wazwaz ldquoThe tanh method and the sine-cosine methodfor solving the KP-MEW equationrdquo International Journal ofComputer Mathematics vol 82 no 2 pp 235ndash246 2005

[27] A Saha ldquoBifurcation of travelling wave solutions for the gener-alized KP-NEW equationsrdquo Communications in Nonlinear Sci-ence and Numerical Simulation vol 17 no 9 pp 3539ndash35512012

[28] MWei S Tang H Fu and G Chen ldquoSingle peak solitary wavesolutions for the generalized KP-MEW (2 2) equation under

Mathematical Problems in Engineering 7

boundary conditionrdquo Applied Mathematics and Computationvol 219 no 17 pp 8979ndash8990 2013

[29] S C Anco and G Bluman ldquoDirect construction method forconservation laws of partial differential equations Part I exam-ples of conservation law classificationsrdquo European Journal ofApplied Mathematics vol 13 no 5 pp 545ndash566 2002

[30] K R Adem and C M Khalique ldquoConservation laws andtraveling wave solutions of a generalized nonlinear ZK-BBMequationrdquo Abstract and Applied Analysis vol 2014 Article ID139513 5 pages 2014

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

Mathematical Problems in Engineering 3

of the KP-MEW equation (1) for arbitrary values of 119899 in thefollowing form

119906 (119909 119910 119905)

= ((119899 + 1)(1198872 minus 120574 (119886 + 119888)

2)

21205731198872)

1(119899minus1)

sech2(119899minus1) [119876]

(14)

where

119876 =radic120574 (119886 + 119888)

2+ 1198872

2radic2(radic2 (1 minus 119899)

119888radic120573119911 + (119899 minus 1) 119862)

119911 = 119905 minus 119909 minus(119886 + 119888)

119887119910

(15)

and 119862 is a constant of integration By taking 119899 = 2 120572 = 12120573 = 1 120574 = 1 119886 = 1 119887 = 1 119888 = 1 119905 = 0 and 119862 = 1 in (14) theprofile of the solution is given in Figure 1

Case 2 Consider the following 1198891198831+ 1198834

The symmetry 1198891198831+1198834gives rise to the three invariants

119891 =119905

1199102 119892 =

119889 ln119910 + 119899119909 minus 119909

119899 minus 1 120601 = 119906119910

2(119899minus1)

(16)

By treating 120601 as the new dependent variable and 119891 and 119892

as new independent variables the KP-MEW equation (1)transforms to

1205731198992120601119891119892119892119892

minus 2120573119899120601119891119892119892119892

+ 2119889120574120601119892119892

+ 2119899120574120601 + 41198912120574120601119891119891

minus 2119891120574120601119891minus 31198861120574120601119892+ 120601119891119892

minus 120572119899120601119899minus2

1206012

119892

+ 1205721198994120601119899minus2

1206012

119892+ 3120572119899

2120601119899minus2

1206012

119892+ 6119891119899

2120574120601119891

minus 4119891119899120574120601119891minus 3120572119899

3120601119899minus2

1206012

119892+ 120572119899120601

119899minus1120601119892119892

+ 1205721198993120601119899minus1

120601119892119892

minus 81198912119899120574120601119891119891

+ 411989121198992120574120601119891119891

+ 4119889119891120574120601119891119892

minus 21205721198992120601119899minus1

120601119892119892

+ 2120574120601 + 120573120601119891119892119892119892

+ 1198911198992120601119892minus 2119891119899120601

119892minus 4119889119891119899120574120601

119891119892minus 119889119891119899120574120601

119892= 0

(17)

Equation (17) has a single Lie point symmetry namely

Γ1=

120597

120597119892 (18)

and this symmetry yields the two invariants

119903 = 119891 119865 = 120601 (19)

which gives rise to a group-invariant solution 119865 = 119865(119903) andconsequently using these invariants (17) is then transformedto a second-order Cauchy-Euler ODE

(211989921199032minus 41198991199032+ 21199032) 11986510158401015840+ (31198992119903 minus 2119899119903 minus 119903) 119865

1015840

+ (1 + 119899) 119865 = 0

(20)

2

2

0

0

0

minus2

minus2

minus2

minus4

minus6

minus8

y

x

u

Figure 1 Profile of solution (14)

Now solving this equation and reverting back to the originalvariables we obtain the following solution of the KP-MEWequation (1)

119906 (119909 119910 119905) =1

1199102(119899minus1)(1198621119903minus(119899+1)2(119899minus1)

+ 1198622119903minus1(119899minus1)

) (21)

where 119903 = 1199051199102 and 1198621and 119862

2are constants of integration

3 (1198661015840119866)-Expansion Method

In this section we use the (1198661015840119866)-expansion method [9 30]to obtain a few exact solutions of the KP-MEW equation (1)for 119899 = 2 and 119899 = 3

Let us consider the solutions of (11) in the form

119865 (119911) =

119872

sum119894=0

A119894(1198661015840 (119911)

119866 (119911))

119894

(22)

where 119866(119911) satisfies

11986610158401015840+ 1205821198661015840+ 120583119866 = 0 (23)

and 120582 and 120583 are constantsThe homogeneous balancemethodbetween the highest order derivative and highest ordernonlinear term appearing in (11) determines the value of 119872andA

0 A

119872are constants to be determined

Consider 119899 = 2 Application of the balancing procedure tofourth-order ODE (11) yields119872 = 2 so the solution of (11) isof the form

119865 (119911) = A0+A1(1198661015840 (119911)

119866 (119911)) +A

2(1198661015840 (119911)

119866 (119911))

2

(24)

4 Mathematical Problems in Engineering

Substituting (23) and (24) into (11) leads to an overdeterminedsystem of algebraic equations Solving this system of algebraicequations with the aid of Maple we obtain

A0=

119887212057212058221198602+ 81198872120583120572119860

2minus 61205741198862 minus 12119886119888120574 minus 61205741198882 + 61198872

121205721198872

A1=

6120573120582

120572

A2=

6120573

120572

(25)

Now using the general solution of (23) in (24) we have thefollowing three types of travelling wave solutions of the KP-MEW equation (1)

When 1205822 minus 4120583 gt 0 we obtain the hyperbolic function

solution

119906 (119909 119910 119905)

= A0+A1(minus

120582

2+ 1205751

1198621sinh (120575

1119911) + 119862

2cosh (120575

1119911)

1198621cosh (120575

1119911) + 119862

2sinh (120575

1119911)

)

+A2(minus

120582

2+ 1205751

1198621sinh (120575

1119911) + 119862

2cosh (120575

1119911)

1198621cosh (120575

1119911) + 119862

2sinh (120575

1119911)

)

2

(26)

where 119911 = 119905 minus 119909 minus ((119886 + 119888)119887)119910 1205751= (12)radic1205822 minus 4120583 and 119862

1

and 1198622are arbitrary constants

The profile of the solution (26) is given in Figure 2When 1205822 minus 4120583 lt 0 we obtain the trigonometric function

solution

119906 (119909 119910 119905)

= A0+A1(minus

120582

2+ 1205752

minus1198621sin (1205752119911) + 119862

2cos (120575

2119911)

1198621cos (120575

2119911) + 119862

2sin (1205752119911)

)

+A2(minus

120582

2+ 1205752

minus1198621sin (1205752119911) + 119862

2cos (120575

2119911)

1198621cos (120575

2119911) + 119862

2sin (1205752119911)

)

2

(27)

where 119911 = 119905 minus 119909 minus ((119886 + 119888)119887)119910 1205752= (12)radic4120583 minus 1205822 and 119862

1

and 1198622are arbitrary constants

The profile of the solution (27) is given in Figure 3When 1205822 minus 4120583 = 0 we obtain the rational function

solution

119906 (119909 119910 119905) = A0+A1(minus

120582

2+

1198622

1198621+ 1198622119911)

+A2(minus

120582

2+

1198622

1198621+ 1198622119911)

2

(28)

where 119911 = 119905 minus 119909 minus ((119886 + 119888)119887)119910 and 1198621and 119862

2are arbitrary

constantsThe profile of the solution (28) is given in Figure 4

2

2

0

0

010

20

30

40

minus2

minus2

y

x

u

Figure 2 Profile of solution (26)

2

2

0

0

500

1000

1500

0

minus2

minus2

y

x

u

Figure 3 Profile of solution (27)

2

2

0

0

0

10

20

minus2

minus2

y

x

u

Figure 4 Profile of solution (28)

Mathematical Problems in Engineering 5

Consider 119899 = 3 Again the application of the balancingprocedure to fourth-order ODE yields119872 = 1 so the solutionof (11) is of the form

119865 (119911) = A0+A1(1198661015840 (119911)

119866 (119911)) (29)

Solving this system of algebraic equations with the aid ofMaple we obtain

120573 =120572

2A2

1

A0=

120582radicminus1198862120574 minus 2119886119888120574 + 1198872 minus 1198882120574

radic12057211988721205822 minus 41205721198872120583

A1=

2radicminus1198862120574 minus 2119886119888120574 + 1198872 minus 1198882120574

radic12057211988721205822 minus 41205721198872120583

(30)

Now using the general solution of (23) in (29) we have thefollowing two types of travelling wave solutions of the KP-MEW equation (1)

When 1205822 minus 4120583 gt 0 we obtain the hyperbolic function

solution

119906 (119909 119910 119905)

= A0+A1(minus

120582

2+ 1205751

1198621sinh (120575

1119911) + 119862

2cosh (120575

1119911)

1198621cosh (120575

1119911) + 119862

2sinh (120575

1119911)

)

(31)

where 119911 = 119905 minus 119909 minus ((119886 + 119888)119887)119910 1205751= (12)radic1205822 minus 4120583 and 119862

1

and 1198622are arbitrary constants

When 1205822 minus 4120583 lt 0 we obtain the trigonometric functionsolution

119906 (119909 119910 119905)

= A0+A1(minus

120582

2+ 1205752

minus1198621sin (1205752119911) + 119862

2cos (120575

2119911)

1198621cos (120575

2119911) + 119862

2sin (1205752119911)

)

(32)

where 119911 = 119905 minus 119909 minus ((119886 + 119888)119887)119910 1205752= (12)radic4120583 minus 1205822 and 119862

1

and 1198622are arbitrary constants

4 Conservation Laws of (1)

In this section we construct conservation laws for (1) Themultiplier method [29 30] will be used

The zeroth-order multiplierΛ(119905 119909 119910 119906) for (1) is given by

Λ = minus1199103

61205741198911015840

1(119905) + 119909119910119891

1(119905) minus

1199102

21205741198911015840

2(119905)

+ 1199091198912(119905) + 119910119891

3(119905) + 119891

4(119905)

(33)

where1198911(119905)1198912(119905)1198913(119905) and119891

4(119905) are arbitrary functions of 119905

Corresponding to the above multiplier we have the followingconserved vectors of (1)

119879119905

1=

1

24120574minus12120574119910119891

1(119905) 119906 minus 6120573120574119910119891

1(119905) 119906119909119909

+ 61205731205741199091199101198911(119905) 119906119909119909119909

+ 121205741199091199101198911(119905) 119906119909

minus12057311991031198911015840

1(119905) 119906119909119909119909

minus 211991031198911015840

1(119905) 119906119909

119879119909

1= minus

119910

241205741199064120572119899119910

21198911015840

1(119905) 119906119909119906119899minus 24120572120574119899119909119891

1(119905) 119906119909119906119899

minus 120573119910211989110158401015840

1(119905) 119906119909119909119906 + 3120573119910

21198911015840

1(119905) 119906119905119909119909

119906

+ 211991021198911015840

1(119905) 119906119905119906 minus 12120573120574119891

1015840

1(119905) 119906119909119906

+ 61205731205741199091198911015840

1(119905) 119906119909119909119906 + 12120573120574119891

1(119905) 119906119905119909119906

minus 181205731205741199091198911(119905) 119906119905119909119909

119906 minus 121205741199091198911(119905) 119906119905119906

+ 241205721205741198911(119905) 119906119899+1

minus 2119910211989110158401015840

1(119905) 1199062

+ 121205741199091198911015840

1(119905) 1199062

119879119910

1=

1

6311991021198911015840

1(119905) 119906 minus 6120574119909119891

1(119905) 119906

+ 61205741199091199101198911(119905) 119906119910minus 11991031198911015840

1(119905) 119906119910

119879119905

2=

1

8120574minus4120574119891

2(119905) 119906 minus 2120573120574119891

2(119905) 119906119909119909

+ 21205731205741199091198912(119905) 119906119909119909119909

+ 41205741199091198912(119905) 119906119909minus 12057311991021198911015840

2(119905) 119906119909119909119909

minus 211991021198911015840

2(119905) 119906119909

119879119909

2= minus

1

81205741199064120572119899119910

21198911015840

2(119905) 119906119909119906119899minus 8120572120574119899119909119891

2(119905) 119906119909119906119899

minus 120573119910211989110158401015840

2119906119909119909119906 + 3120573119910

21198911015840

2119906119905119909119909

119906

+ 211991021198911015840

2(119905) 119906119905119906 minus 4120573120574119891

1015840

2(119905) 119906119909119906

+ 21205731205741199091198911015840

2(119905) 119906119909119909119906 + 4120573120574119891

2(119905) 119906119905119909119906

minus 61205731205741199091198912(119905) 119906119905119909119909

119906 minus 41205741199091198912(119905) 119906119905119906

+ 81205721205741198912(119905) 119906119899+1

minus 2119910211989110158401015840

2(119905) 1199062

+ 41205741199091198911015840

2(119905) 1199062

119879119910

2=

1

221199101198911015840

2(119905) 119906 + 2120574119909119891

2(119905) 119906119910minus 11991021198911015840

2(119905) 119906119910

119879119905

3=

1

41205731199101198913(119905) 119906119909119909119909

+ 21199101198913(119905) 119906119909

119879119909

3= minus

119910

4119906minus4120572119899119891

3(119905) 119906119909119906119899+ 1205731198911015840

3(119905) 119906119909119909119906 minus 3120573119891

3(119905) 119906119905119909119909

119906

minus 21198913(119905) 119906119905119906 + 2119891

1015840

3(119905) 1199062

119879119910

3= 120574119910119891

3(119905) 119906119910minus 1205741198913(119905) 119906

6 Mathematical Problems in Engineering

119879119905

4=

1

41205731198914(119905) 119906119909119909119909

+ 21198914(119905) 119906119909

119879119909

4= minus

1

4119906minus4120572119899119891

4(119905) 119906119909119906119899+ 1205731198911015840

4(119905) 119906119909119909119906 minus 3120573119891

4(119905) 119906119905119909119909

119906

minus 21198914(119905) 119906119905119906 + 2119891

1015840

4(119905) 1199062

119879119910

4= 1205741198914(119905) 119906119910

(34)

Remark The presence of the arbitrary functions in themultiplier leads to a family of infinitely many conservationlaws for (1)

5 Concluding Remarks

In this paper we obtained the solutions of a gener-alized two-dimensional nonlinear Kadomtsev-Petviashvili-modified equal width equation by employing the Lie groupanalysis the optimal systems of one-dimensional subal-gebras and the (119866

1015840119866)-expansion method The solutionsobtained are solitary waves and nontopological solitons Theconservation laws for the underlying equation were alsoderived by using the multiplier method

Conflict of Interests

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

References

[1] M J Ablowitz and P A Clarkson Soliton Nonlinear EvolutionEquations and Inverse Scattering Cambridge University PressCambridge UK 1991

[2] M Wang Y Zhou and Z Li ldquoApplication of a homogeneousbalance method to exact solutions of nonlinear equations inmathematical physicsrdquo Physics Letters A vol 216 no 1ndash5 pp67ndash75 1996

[3] C H Gu Soliton Theory and Its Application Zhejiang Scienceand Technology Press Zhejiang China 1990

[4] Y Chen and Z Yan ldquoThe Weierstrass elliptic function expan-sion method and its applications in nonlinear wave equationsrdquoChaos Solitons and Fractals vol 29 no 4 pp 948ndash964 2006

[5] V B Matveev and M A Salle Darboux Transformation andSoliton Springer Berlin Germany 1991

[6] J Hu ldquoExplicit solutions to three nonlinear physical modelsrdquoPhysics Letters A vol 287 no 1-2 pp 81ndash89 2001

[7] J Hu and H Zhang ldquoA new method for finding exact travelingwave solutions to nonlinear partial differential equationsrdquoPhysics Letters A vol 286 no 2-3 pp 175ndash179 2001

[8] R Hirota The Direct Method in Soliton Theory vol 155 ofCambridge Tracts in Mathematics Cambridge University PressCambridge UK 2004

[9] M Wang X Li and J Zhang ldquoThe (1198661015840

119866)-expansion methodand travelling wave solutions of nonlinear evolution equationsin mathematical physicsrdquo Physics Letters A vol 372 no 4 pp417ndash423 2008

[10] D Lu ldquoJacobi elliptic function solutions for two variant Boussi-nesq equationsrdquo Chaos Solitons amp Fractals vol 24 no 5 pp1373ndash1385 2005

[11] Z Yan ldquoAbundant families of Jacobi elliptic function solutionsof the (2 + 1)-dimensional integrable Davey-Stewartson-typeequation via a new methodrdquo Chaos Solitons and Fractals vol18 no 2 pp 299ndash309 2003

[12] S-Y Lou and J Lu ldquoSpecial solutions from the variableseparation approach the Davey-Stewartson equationrdquo Journalof Physics A Mathematical and General vol 29 no 14 pp4209ndash4215 1996

[13] A-M Wazwaz ldquoThe tanh and the sine-cosine methods forcompact and noncompact solutions of the nonlinear Klein-Gordon equationrdquo Applied Mathematics and Computation vol167 no 2 pp 1179ndash1195 2005

[14] Z Yan ldquoThe new tri-function method to multiple exact solu-tions of nonlinear wave equationsrdquo Physica Scripta vol 78 no3 Article ID 035001 5 pages 2008

[15] Z Yan ldquoPeriodic solitary and rational wave solutions of the3D extended quantum Zakharov-Kuznetsov equation in densequantum plasmasrdquo Physics Letters Section A General Atomicand Solid State Physics vol 373 no 29 pp 2432ndash2437 2009

[16] M Wang and X Li ldquoExtended 119865-expansion method and peri-odic wave solutions for the generalized Zakharov equationsrdquoPhysics Letters A vol 343 no 1ndash3 pp 48ndash54 2005

[17] S Zhang ldquoApplication of Exp-function method to high-dimensional nonlinear evolution equationrdquo Chaos Solitons andFractals vol 38 no 1 pp 270ndash276 2008

[18] W-X Ma T Huang and Y Zhang ldquoA multiple exp-functionmethod for nonlinear differential equations and its applicationrdquoPhysica Scripta vol 82 no 6 Article ID 065003 2010

[19] G W Bluman and S Kumei Symmetries and DifferentialEquations AppliedMathematical Sciences Springer NewYorkNY USA 1989

[20] P J Olver Applications of Lie Groups to Differential Equationsvol 107 of Graduate Texts in Mathematics Springer New YorkNY USA 2nd edition 1993

[21] L V Ovsiannikov Group Analysis of Differential EquationsAcademic Press New York NY USA 1982

[22] N H Ibragimov CRC Handbook of Lie Group Analysis ofDifferential Equations vol 1 CRC Press Boca Raton Fla USA1994

[23] N H Ibragimov CRC Handbook of Lie Group Analysis ofDifferential Equations vol 2 CRC Press Boca Raton Fla USA1995

[24] N H Ibragimov CRC Handbook of Lie Group Analysis ofDifferential Equations vol 3 CRC Press Boca Raton Fla USA1996

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

[26] A-M Wazwaz ldquoThe tanh method and the sine-cosine methodfor solving the KP-MEW equationrdquo International Journal ofComputer Mathematics vol 82 no 2 pp 235ndash246 2005

[27] A Saha ldquoBifurcation of travelling wave solutions for the gener-alized KP-NEW equationsrdquo Communications in Nonlinear Sci-ence and Numerical Simulation vol 17 no 9 pp 3539ndash35512012

[28] MWei S Tang H Fu and G Chen ldquoSingle peak solitary wavesolutions for the generalized KP-MEW (2 2) equation under

Mathematical Problems in Engineering 7

boundary conditionrdquo Applied Mathematics and Computationvol 219 no 17 pp 8979ndash8990 2013

[29] S C Anco and G Bluman ldquoDirect construction method forconservation laws of partial differential equations Part I exam-ples of conservation law classificationsrdquo European Journal ofApplied Mathematics vol 13 no 5 pp 545ndash566 2002

[30] K R Adem and C M Khalique ldquoConservation laws andtraveling wave solutions of a generalized nonlinear ZK-BBMequationrdquo Abstract and Applied Analysis vol 2014 Article ID139513 5 pages 2014

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 Mathematical Problems in Engineering

Substituting (23) and (24) into (11) leads to an overdeterminedsystem of algebraic equations Solving this system of algebraicequations with the aid of Maple we obtain

A0=

119887212057212058221198602+ 81198872120583120572119860

2minus 61205741198862 minus 12119886119888120574 minus 61205741198882 + 61198872

121205721198872

A1=

6120573120582

120572

A2=

6120573

120572

(25)

Now using the general solution of (23) in (24) we have thefollowing three types of travelling wave solutions of the KP-MEW equation (1)

When 1205822 minus 4120583 gt 0 we obtain the hyperbolic function

solution

119906 (119909 119910 119905)

= A0+A1(minus

120582

2+ 1205751

1198621sinh (120575

1119911) + 119862

2cosh (120575

1119911)

1198621cosh (120575

1119911) + 119862

2sinh (120575

1119911)

)

+A2(minus

120582

2+ 1205751

1198621sinh (120575

1119911) + 119862

2cosh (120575

1119911)

1198621cosh (120575

1119911) + 119862

2sinh (120575

1119911)

)

2

(26)

where 119911 = 119905 minus 119909 minus ((119886 + 119888)119887)119910 1205751= (12)radic1205822 minus 4120583 and 119862

1

and 1198622are arbitrary constants

The profile of the solution (26) is given in Figure 2When 1205822 minus 4120583 lt 0 we obtain the trigonometric function

solution

119906 (119909 119910 119905)

= A0+A1(minus

120582

2+ 1205752

minus1198621sin (1205752119911) + 119862

2cos (120575

2119911)

1198621cos (120575

2119911) + 119862

2sin (1205752119911)

)

+A2(minus

120582

2+ 1205752

minus1198621sin (1205752119911) + 119862

2cos (120575

2119911)

1198621cos (120575

2119911) + 119862

2sin (1205752119911)

)

2

(27)

where 119911 = 119905 minus 119909 minus ((119886 + 119888)119887)119910 1205752= (12)radic4120583 minus 1205822 and 119862

1

and 1198622are arbitrary constants

The profile of the solution (27) is given in Figure 3When 1205822 minus 4120583 = 0 we obtain the rational function

solution

119906 (119909 119910 119905) = A0+A1(minus

120582

2+

1198622

1198621+ 1198622119911)

+A2(minus

120582

2+

1198622

1198621+ 1198622119911)

2

(28)

where 119911 = 119905 minus 119909 minus ((119886 + 119888)119887)119910 and 1198621and 119862

2are arbitrary

constantsThe profile of the solution (28) is given in Figure 4

2

2

0

0

010

20

30

40

minus2

minus2

y

x

u

Figure 2 Profile of solution (26)

2

2

0

0

500

1000

1500

0

minus2

minus2

y

x

u

Figure 3 Profile of solution (27)

2

2

0

0

0

10

20

minus2

minus2

y

x

u

Figure 4 Profile of solution (28)

Mathematical Problems in Engineering 5

Consider 119899 = 3 Again the application of the balancingprocedure to fourth-order ODE yields119872 = 1 so the solutionof (11) is of the form

119865 (119911) = A0+A1(1198661015840 (119911)

119866 (119911)) (29)

Solving this system of algebraic equations with the aid ofMaple we obtain

120573 =120572

2A2

1

A0=

120582radicminus1198862120574 minus 2119886119888120574 + 1198872 minus 1198882120574

radic12057211988721205822 minus 41205721198872120583

A1=

2radicminus1198862120574 minus 2119886119888120574 + 1198872 minus 1198882120574

radic12057211988721205822 minus 41205721198872120583

(30)

Now using the general solution of (23) in (29) we have thefollowing two types of travelling wave solutions of the KP-MEW equation (1)

When 1205822 minus 4120583 gt 0 we obtain the hyperbolic function

solution

119906 (119909 119910 119905)

= A0+A1(minus

120582

2+ 1205751

1198621sinh (120575

1119911) + 119862

2cosh (120575

1119911)

1198621cosh (120575

1119911) + 119862

2sinh (120575

1119911)

)

(31)

where 119911 = 119905 minus 119909 minus ((119886 + 119888)119887)119910 1205751= (12)radic1205822 minus 4120583 and 119862

1

and 1198622are arbitrary constants

When 1205822 minus 4120583 lt 0 we obtain the trigonometric functionsolution

119906 (119909 119910 119905)

= A0+A1(minus

120582

2+ 1205752

minus1198621sin (1205752119911) + 119862

2cos (120575

2119911)

1198621cos (120575

2119911) + 119862

2sin (1205752119911)

)

(32)

where 119911 = 119905 minus 119909 minus ((119886 + 119888)119887)119910 1205752= (12)radic4120583 minus 1205822 and 119862

1

and 1198622are arbitrary constants

4 Conservation Laws of (1)

In this section we construct conservation laws for (1) Themultiplier method [29 30] will be used

The zeroth-order multiplierΛ(119905 119909 119910 119906) for (1) is given by

Λ = minus1199103

61205741198911015840

1(119905) + 119909119910119891

1(119905) minus

1199102

21205741198911015840

2(119905)

+ 1199091198912(119905) + 119910119891

3(119905) + 119891

4(119905)

(33)

where1198911(119905)1198912(119905)1198913(119905) and119891

4(119905) are arbitrary functions of 119905

Corresponding to the above multiplier we have the followingconserved vectors of (1)

119879119905

1=

1

24120574minus12120574119910119891

1(119905) 119906 minus 6120573120574119910119891

1(119905) 119906119909119909

+ 61205731205741199091199101198911(119905) 119906119909119909119909

+ 121205741199091199101198911(119905) 119906119909

minus12057311991031198911015840

1(119905) 119906119909119909119909

minus 211991031198911015840

1(119905) 119906119909

119879119909

1= minus

119910

241205741199064120572119899119910

21198911015840

1(119905) 119906119909119906119899minus 24120572120574119899119909119891

1(119905) 119906119909119906119899

minus 120573119910211989110158401015840

1(119905) 119906119909119909119906 + 3120573119910

21198911015840

1(119905) 119906119905119909119909

119906

+ 211991021198911015840

1(119905) 119906119905119906 minus 12120573120574119891

1015840

1(119905) 119906119909119906

+ 61205731205741199091198911015840

1(119905) 119906119909119909119906 + 12120573120574119891

1(119905) 119906119905119909119906

minus 181205731205741199091198911(119905) 119906119905119909119909

119906 minus 121205741199091198911(119905) 119906119905119906

+ 241205721205741198911(119905) 119906119899+1

minus 2119910211989110158401015840

1(119905) 1199062

+ 121205741199091198911015840

1(119905) 1199062

119879119910

1=

1

6311991021198911015840

1(119905) 119906 minus 6120574119909119891

1(119905) 119906

+ 61205741199091199101198911(119905) 119906119910minus 11991031198911015840

1(119905) 119906119910

119879119905

2=

1

8120574minus4120574119891

2(119905) 119906 minus 2120573120574119891

2(119905) 119906119909119909

+ 21205731205741199091198912(119905) 119906119909119909119909

+ 41205741199091198912(119905) 119906119909minus 12057311991021198911015840

2(119905) 119906119909119909119909

minus 211991021198911015840

2(119905) 119906119909

119879119909

2= minus

1

81205741199064120572119899119910

21198911015840

2(119905) 119906119909119906119899minus 8120572120574119899119909119891

2(119905) 119906119909119906119899

minus 120573119910211989110158401015840

2119906119909119909119906 + 3120573119910

21198911015840

2119906119905119909119909

119906

+ 211991021198911015840

2(119905) 119906119905119906 minus 4120573120574119891

1015840

2(119905) 119906119909119906

+ 21205731205741199091198911015840

2(119905) 119906119909119909119906 + 4120573120574119891

2(119905) 119906119905119909119906

minus 61205731205741199091198912(119905) 119906119905119909119909

119906 minus 41205741199091198912(119905) 119906119905119906

+ 81205721205741198912(119905) 119906119899+1

minus 2119910211989110158401015840

2(119905) 1199062

+ 41205741199091198911015840

2(119905) 1199062

119879119910

2=

1

221199101198911015840

2(119905) 119906 + 2120574119909119891

2(119905) 119906119910minus 11991021198911015840

2(119905) 119906119910

119879119905

3=

1

41205731199101198913(119905) 119906119909119909119909

+ 21199101198913(119905) 119906119909

119879119909

3= minus

119910

4119906minus4120572119899119891

3(119905) 119906119909119906119899+ 1205731198911015840

3(119905) 119906119909119909119906 minus 3120573119891

3(119905) 119906119905119909119909

119906

minus 21198913(119905) 119906119905119906 + 2119891

1015840

3(119905) 1199062

119879119910

3= 120574119910119891

3(119905) 119906119910minus 1205741198913(119905) 119906

6 Mathematical Problems in Engineering

119879119905

4=

1

41205731198914(119905) 119906119909119909119909

+ 21198914(119905) 119906119909

119879119909

4= minus

1

4119906minus4120572119899119891

4(119905) 119906119909119906119899+ 1205731198911015840

4(119905) 119906119909119909119906 minus 3120573119891

4(119905) 119906119905119909119909

119906

minus 21198914(119905) 119906119905119906 + 2119891

1015840

4(119905) 1199062

119879119910

4= 1205741198914(119905) 119906119910

(34)

Remark The presence of the arbitrary functions in themultiplier leads to a family of infinitely many conservationlaws for (1)

5 Concluding Remarks

In this paper we obtained the solutions of a gener-alized two-dimensional nonlinear Kadomtsev-Petviashvili-modified equal width equation by employing the Lie groupanalysis the optimal systems of one-dimensional subal-gebras and the (119866

1015840119866)-expansion method The solutionsobtained are solitary waves and nontopological solitons Theconservation laws for the underlying equation were alsoderived by using the multiplier method

Conflict of Interests

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

References

[1] M J Ablowitz and P A Clarkson Soliton Nonlinear EvolutionEquations and Inverse Scattering Cambridge University PressCambridge UK 1991

[2] M Wang Y Zhou and Z Li ldquoApplication of a homogeneousbalance method to exact solutions of nonlinear equations inmathematical physicsrdquo Physics Letters A vol 216 no 1ndash5 pp67ndash75 1996

[3] C H Gu Soliton Theory and Its Application Zhejiang Scienceand Technology Press Zhejiang China 1990

[4] Y Chen and Z Yan ldquoThe Weierstrass elliptic function expan-sion method and its applications in nonlinear wave equationsrdquoChaos Solitons and Fractals vol 29 no 4 pp 948ndash964 2006

[5] V B Matveev and M A Salle Darboux Transformation andSoliton Springer Berlin Germany 1991

[6] J Hu ldquoExplicit solutions to three nonlinear physical modelsrdquoPhysics Letters A vol 287 no 1-2 pp 81ndash89 2001

[7] J Hu and H Zhang ldquoA new method for finding exact travelingwave solutions to nonlinear partial differential equationsrdquoPhysics Letters A vol 286 no 2-3 pp 175ndash179 2001

[8] R Hirota The Direct Method in Soliton Theory vol 155 ofCambridge Tracts in Mathematics Cambridge University PressCambridge UK 2004

[9] M Wang X Li and J Zhang ldquoThe (1198661015840

119866)-expansion methodand travelling wave solutions of nonlinear evolution equationsin mathematical physicsrdquo Physics Letters A vol 372 no 4 pp417ndash423 2008

[10] D Lu ldquoJacobi elliptic function solutions for two variant Boussi-nesq equationsrdquo Chaos Solitons amp Fractals vol 24 no 5 pp1373ndash1385 2005

[11] Z Yan ldquoAbundant families of Jacobi elliptic function solutionsof the (2 + 1)-dimensional integrable Davey-Stewartson-typeequation via a new methodrdquo Chaos Solitons and Fractals vol18 no 2 pp 299ndash309 2003

[12] S-Y Lou and J Lu ldquoSpecial solutions from the variableseparation approach the Davey-Stewartson equationrdquo Journalof Physics A Mathematical and General vol 29 no 14 pp4209ndash4215 1996

[13] A-M Wazwaz ldquoThe tanh and the sine-cosine methods forcompact and noncompact solutions of the nonlinear Klein-Gordon equationrdquo Applied Mathematics and Computation vol167 no 2 pp 1179ndash1195 2005

[14] Z Yan ldquoThe new tri-function method to multiple exact solu-tions of nonlinear wave equationsrdquo Physica Scripta vol 78 no3 Article ID 035001 5 pages 2008

[15] Z Yan ldquoPeriodic solitary and rational wave solutions of the3D extended quantum Zakharov-Kuznetsov equation in densequantum plasmasrdquo Physics Letters Section A General Atomicand Solid State Physics vol 373 no 29 pp 2432ndash2437 2009

[16] M Wang and X Li ldquoExtended 119865-expansion method and peri-odic wave solutions for the generalized Zakharov equationsrdquoPhysics Letters A vol 343 no 1ndash3 pp 48ndash54 2005

[17] S Zhang ldquoApplication of Exp-function method to high-dimensional nonlinear evolution equationrdquo Chaos Solitons andFractals vol 38 no 1 pp 270ndash276 2008

[18] W-X Ma T Huang and Y Zhang ldquoA multiple exp-functionmethod for nonlinear differential equations and its applicationrdquoPhysica Scripta vol 82 no 6 Article ID 065003 2010

[19] G W Bluman and S Kumei Symmetries and DifferentialEquations AppliedMathematical Sciences Springer NewYorkNY USA 1989

[20] P J Olver Applications of Lie Groups to Differential Equationsvol 107 of Graduate Texts in Mathematics Springer New YorkNY USA 2nd edition 1993

[21] L V Ovsiannikov Group Analysis of Differential EquationsAcademic Press New York NY USA 1982

[22] N H Ibragimov CRC Handbook of Lie Group Analysis ofDifferential Equations vol 1 CRC Press Boca Raton Fla USA1994

[23] N H Ibragimov CRC Handbook of Lie Group Analysis ofDifferential Equations vol 2 CRC Press Boca Raton Fla USA1995

[24] N H Ibragimov CRC Handbook of Lie Group Analysis ofDifferential Equations vol 3 CRC Press Boca Raton Fla USA1996

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

[26] A-M Wazwaz ldquoThe tanh method and the sine-cosine methodfor solving the KP-MEW equationrdquo International Journal ofComputer Mathematics vol 82 no 2 pp 235ndash246 2005

[27] A Saha ldquoBifurcation of travelling wave solutions for the gener-alized KP-NEW equationsrdquo Communications in Nonlinear Sci-ence and Numerical Simulation vol 17 no 9 pp 3539ndash35512012

[28] MWei S Tang H Fu and G Chen ldquoSingle peak solitary wavesolutions for the generalized KP-MEW (2 2) equation under

Mathematical Problems in Engineering 7

boundary conditionrdquo Applied Mathematics and Computationvol 219 no 17 pp 8979ndash8990 2013

[29] S C Anco and G Bluman ldquoDirect construction method forconservation laws of partial differential equations Part I exam-ples of conservation law classificationsrdquo European Journal ofApplied Mathematics vol 13 no 5 pp 545ndash566 2002

[30] K R Adem and C M Khalique ldquoConservation laws andtraveling wave solutions of a generalized nonlinear ZK-BBMequationrdquo Abstract and Applied Analysis vol 2014 Article ID139513 5 pages 2014

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

Mathematical Problems in Engineering 5

Consider 119899 = 3 Again the application of the balancingprocedure to fourth-order ODE yields119872 = 1 so the solutionof (11) is of the form

119865 (119911) = A0+A1(1198661015840 (119911)

119866 (119911)) (29)

Solving this system of algebraic equations with the aid ofMaple we obtain

120573 =120572

2A2

1

A0=

120582radicminus1198862120574 minus 2119886119888120574 + 1198872 minus 1198882120574

radic12057211988721205822 minus 41205721198872120583

A1=

2radicminus1198862120574 minus 2119886119888120574 + 1198872 minus 1198882120574

radic12057211988721205822 minus 41205721198872120583

(30)

Now using the general solution of (23) in (29) we have thefollowing two types of travelling wave solutions of the KP-MEW equation (1)

When 1205822 minus 4120583 gt 0 we obtain the hyperbolic function

solution

119906 (119909 119910 119905)

= A0+A1(minus

120582

2+ 1205751

1198621sinh (120575

1119911) + 119862

2cosh (120575

1119911)

1198621cosh (120575

1119911) + 119862

2sinh (120575

1119911)

)

(31)

where 119911 = 119905 minus 119909 minus ((119886 + 119888)119887)119910 1205751= (12)radic1205822 minus 4120583 and 119862

1

and 1198622are arbitrary constants

When 1205822 minus 4120583 lt 0 we obtain the trigonometric functionsolution

119906 (119909 119910 119905)

= A0+A1(minus

120582

2+ 1205752

minus1198621sin (1205752119911) + 119862

2cos (120575

2119911)

1198621cos (120575

2119911) + 119862

2sin (1205752119911)

)

(32)

where 119911 = 119905 minus 119909 minus ((119886 + 119888)119887)119910 1205752= (12)radic4120583 minus 1205822 and 119862

1

and 1198622are arbitrary constants

4 Conservation Laws of (1)

In this section we construct conservation laws for (1) Themultiplier method [29 30] will be used

The zeroth-order multiplierΛ(119905 119909 119910 119906) for (1) is given by

Λ = minus1199103

61205741198911015840

1(119905) + 119909119910119891

1(119905) minus

1199102

21205741198911015840

2(119905)

+ 1199091198912(119905) + 119910119891

3(119905) + 119891

4(119905)

(33)

where1198911(119905)1198912(119905)1198913(119905) and119891

4(119905) are arbitrary functions of 119905

Corresponding to the above multiplier we have the followingconserved vectors of (1)

119879119905

1=

1

24120574minus12120574119910119891

1(119905) 119906 minus 6120573120574119910119891

1(119905) 119906119909119909

+ 61205731205741199091199101198911(119905) 119906119909119909119909

+ 121205741199091199101198911(119905) 119906119909

minus12057311991031198911015840

1(119905) 119906119909119909119909

minus 211991031198911015840

1(119905) 119906119909

119879119909

1= minus

119910

241205741199064120572119899119910

21198911015840

1(119905) 119906119909119906119899minus 24120572120574119899119909119891

1(119905) 119906119909119906119899

minus 120573119910211989110158401015840

1(119905) 119906119909119909119906 + 3120573119910

21198911015840

1(119905) 119906119905119909119909

119906

+ 211991021198911015840

1(119905) 119906119905119906 minus 12120573120574119891

1015840

1(119905) 119906119909119906

+ 61205731205741199091198911015840

1(119905) 119906119909119909119906 + 12120573120574119891

1(119905) 119906119905119909119906

minus 181205731205741199091198911(119905) 119906119905119909119909

119906 minus 121205741199091198911(119905) 119906119905119906

+ 241205721205741198911(119905) 119906119899+1

minus 2119910211989110158401015840

1(119905) 1199062

+ 121205741199091198911015840

1(119905) 1199062

119879119910

1=

1

6311991021198911015840

1(119905) 119906 minus 6120574119909119891

1(119905) 119906

+ 61205741199091199101198911(119905) 119906119910minus 11991031198911015840

1(119905) 119906119910

119879119905

2=

1

8120574minus4120574119891

2(119905) 119906 minus 2120573120574119891

2(119905) 119906119909119909

+ 21205731205741199091198912(119905) 119906119909119909119909

+ 41205741199091198912(119905) 119906119909minus 12057311991021198911015840

2(119905) 119906119909119909119909

minus 211991021198911015840

2(119905) 119906119909

119879119909

2= minus

1

81205741199064120572119899119910

21198911015840

2(119905) 119906119909119906119899minus 8120572120574119899119909119891

2(119905) 119906119909119906119899

minus 120573119910211989110158401015840

2119906119909119909119906 + 3120573119910

21198911015840

2119906119905119909119909

119906

+ 211991021198911015840

2(119905) 119906119905119906 minus 4120573120574119891

1015840

2(119905) 119906119909119906

+ 21205731205741199091198911015840

2(119905) 119906119909119909119906 + 4120573120574119891

2(119905) 119906119905119909119906

minus 61205731205741199091198912(119905) 119906119905119909119909

119906 minus 41205741199091198912(119905) 119906119905119906

+ 81205721205741198912(119905) 119906119899+1

minus 2119910211989110158401015840

2(119905) 1199062

+ 41205741199091198911015840

2(119905) 1199062

119879119910

2=

1

221199101198911015840

2(119905) 119906 + 2120574119909119891

2(119905) 119906119910minus 11991021198911015840

2(119905) 119906119910

119879119905

3=

1

41205731199101198913(119905) 119906119909119909119909

+ 21199101198913(119905) 119906119909

119879119909

3= minus

119910

4119906minus4120572119899119891

3(119905) 119906119909119906119899+ 1205731198911015840

3(119905) 119906119909119909119906 minus 3120573119891

3(119905) 119906119905119909119909

119906

minus 21198913(119905) 119906119905119906 + 2119891

1015840

3(119905) 1199062

119879119910

3= 120574119910119891

3(119905) 119906119910minus 1205741198913(119905) 119906

6 Mathematical Problems in Engineering

119879119905

4=

1

41205731198914(119905) 119906119909119909119909

+ 21198914(119905) 119906119909

119879119909

4= minus

1

4119906minus4120572119899119891

4(119905) 119906119909119906119899+ 1205731198911015840

4(119905) 119906119909119909119906 minus 3120573119891

4(119905) 119906119905119909119909

119906

minus 21198914(119905) 119906119905119906 + 2119891

1015840

4(119905) 1199062

119879119910

4= 1205741198914(119905) 119906119910

(34)

Remark The presence of the arbitrary functions in themultiplier leads to a family of infinitely many conservationlaws for (1)

5 Concluding Remarks

In this paper we obtained the solutions of a gener-alized two-dimensional nonlinear Kadomtsev-Petviashvili-modified equal width equation by employing the Lie groupanalysis the optimal systems of one-dimensional subal-gebras and the (119866

1015840119866)-expansion method The solutionsobtained are solitary waves and nontopological solitons Theconservation laws for the underlying equation were alsoderived by using the multiplier method

Conflict of Interests

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

References

[1] M J Ablowitz and P A Clarkson Soliton Nonlinear EvolutionEquations and Inverse Scattering Cambridge University PressCambridge UK 1991

[2] M Wang Y Zhou and Z Li ldquoApplication of a homogeneousbalance method to exact solutions of nonlinear equations inmathematical physicsrdquo Physics Letters A vol 216 no 1ndash5 pp67ndash75 1996

[3] C H Gu Soliton Theory and Its Application Zhejiang Scienceand Technology Press Zhejiang China 1990

[4] Y Chen and Z Yan ldquoThe Weierstrass elliptic function expan-sion method and its applications in nonlinear wave equationsrdquoChaos Solitons and Fractals vol 29 no 4 pp 948ndash964 2006

[5] V B Matveev and M A Salle Darboux Transformation andSoliton Springer Berlin Germany 1991

[6] J Hu ldquoExplicit solutions to three nonlinear physical modelsrdquoPhysics Letters A vol 287 no 1-2 pp 81ndash89 2001

[7] J Hu and H Zhang ldquoA new method for finding exact travelingwave solutions to nonlinear partial differential equationsrdquoPhysics Letters A vol 286 no 2-3 pp 175ndash179 2001

[8] R Hirota The Direct Method in Soliton Theory vol 155 ofCambridge Tracts in Mathematics Cambridge University PressCambridge UK 2004

[9] M Wang X Li and J Zhang ldquoThe (1198661015840

119866)-expansion methodand travelling wave solutions of nonlinear evolution equationsin mathematical physicsrdquo Physics Letters A vol 372 no 4 pp417ndash423 2008

[10] D Lu ldquoJacobi elliptic function solutions for two variant Boussi-nesq equationsrdquo Chaos Solitons amp Fractals vol 24 no 5 pp1373ndash1385 2005

[11] Z Yan ldquoAbundant families of Jacobi elliptic function solutionsof the (2 + 1)-dimensional integrable Davey-Stewartson-typeequation via a new methodrdquo Chaos Solitons and Fractals vol18 no 2 pp 299ndash309 2003

[12] S-Y Lou and J Lu ldquoSpecial solutions from the variableseparation approach the Davey-Stewartson equationrdquo Journalof Physics A Mathematical and General vol 29 no 14 pp4209ndash4215 1996

[13] A-M Wazwaz ldquoThe tanh and the sine-cosine methods forcompact and noncompact solutions of the nonlinear Klein-Gordon equationrdquo Applied Mathematics and Computation vol167 no 2 pp 1179ndash1195 2005

[14] Z Yan ldquoThe new tri-function method to multiple exact solu-tions of nonlinear wave equationsrdquo Physica Scripta vol 78 no3 Article ID 035001 5 pages 2008

[15] Z Yan ldquoPeriodic solitary and rational wave solutions of the3D extended quantum Zakharov-Kuznetsov equation in densequantum plasmasrdquo Physics Letters Section A General Atomicand Solid State Physics vol 373 no 29 pp 2432ndash2437 2009

[16] M Wang and X Li ldquoExtended 119865-expansion method and peri-odic wave solutions for the generalized Zakharov equationsrdquoPhysics Letters A vol 343 no 1ndash3 pp 48ndash54 2005

[17] S Zhang ldquoApplication of Exp-function method to high-dimensional nonlinear evolution equationrdquo Chaos Solitons andFractals vol 38 no 1 pp 270ndash276 2008

[18] W-X Ma T Huang and Y Zhang ldquoA multiple exp-functionmethod for nonlinear differential equations and its applicationrdquoPhysica Scripta vol 82 no 6 Article ID 065003 2010

[19] G W Bluman and S Kumei Symmetries and DifferentialEquations AppliedMathematical Sciences Springer NewYorkNY USA 1989

[20] P J Olver Applications of Lie Groups to Differential Equationsvol 107 of Graduate Texts in Mathematics Springer New YorkNY USA 2nd edition 1993

[21] L V Ovsiannikov Group Analysis of Differential EquationsAcademic Press New York NY USA 1982

[22] N H Ibragimov CRC Handbook of Lie Group Analysis ofDifferential Equations vol 1 CRC Press Boca Raton Fla USA1994

[23] N H Ibragimov CRC Handbook of Lie Group Analysis ofDifferential Equations vol 2 CRC Press Boca Raton Fla USA1995

[24] N H Ibragimov CRC Handbook of Lie Group Analysis ofDifferential Equations vol 3 CRC Press Boca Raton Fla USA1996

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

[26] A-M Wazwaz ldquoThe tanh method and the sine-cosine methodfor solving the KP-MEW equationrdquo International Journal ofComputer Mathematics vol 82 no 2 pp 235ndash246 2005

[27] A Saha ldquoBifurcation of travelling wave solutions for the gener-alized KP-NEW equationsrdquo Communications in Nonlinear Sci-ence and Numerical Simulation vol 17 no 9 pp 3539ndash35512012

[28] MWei S Tang H Fu and G Chen ldquoSingle peak solitary wavesolutions for the generalized KP-MEW (2 2) equation under

Mathematical Problems in Engineering 7

boundary conditionrdquo Applied Mathematics and Computationvol 219 no 17 pp 8979ndash8990 2013

[29] S C Anco and G Bluman ldquoDirect construction method forconservation laws of partial differential equations Part I exam-ples of conservation law classificationsrdquo European Journal ofApplied Mathematics vol 13 no 5 pp 545ndash566 2002

[30] K R Adem and C M Khalique ldquoConservation laws andtraveling wave solutions of a generalized nonlinear ZK-BBMequationrdquo Abstract and Applied Analysis vol 2014 Article ID139513 5 pages 2014

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 Mathematical Problems in Engineering

119879119905

4=

1

41205731198914(119905) 119906119909119909119909

+ 21198914(119905) 119906119909

119879119909

4= minus

1

4119906minus4120572119899119891

4(119905) 119906119909119906119899+ 1205731198911015840

4(119905) 119906119909119909119906 minus 3120573119891

4(119905) 119906119905119909119909

119906

minus 21198914(119905) 119906119905119906 + 2119891

1015840

4(119905) 1199062

119879119910

4= 1205741198914(119905) 119906119910

(34)

Remark The presence of the arbitrary functions in themultiplier leads to a family of infinitely many conservationlaws for (1)

5 Concluding Remarks

In this paper we obtained the solutions of a gener-alized two-dimensional nonlinear Kadomtsev-Petviashvili-modified equal width equation by employing the Lie groupanalysis the optimal systems of one-dimensional subal-gebras and the (119866

1015840119866)-expansion method The solutionsobtained are solitary waves and nontopological solitons Theconservation laws for the underlying equation were alsoderived by using the multiplier method

Conflict of Interests

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

References

[1] M J Ablowitz and P A Clarkson Soliton Nonlinear EvolutionEquations and Inverse Scattering Cambridge University PressCambridge UK 1991

[2] M Wang Y Zhou and Z Li ldquoApplication of a homogeneousbalance method to exact solutions of nonlinear equations inmathematical physicsrdquo Physics Letters A vol 216 no 1ndash5 pp67ndash75 1996

[3] C H Gu Soliton Theory and Its Application Zhejiang Scienceand Technology Press Zhejiang China 1990

[4] Y Chen and Z Yan ldquoThe Weierstrass elliptic function expan-sion method and its applications in nonlinear wave equationsrdquoChaos Solitons and Fractals vol 29 no 4 pp 948ndash964 2006

[5] V B Matveev and M A Salle Darboux Transformation andSoliton Springer Berlin Germany 1991

[6] J Hu ldquoExplicit solutions to three nonlinear physical modelsrdquoPhysics Letters A vol 287 no 1-2 pp 81ndash89 2001

[7] J Hu and H Zhang ldquoA new method for finding exact travelingwave solutions to nonlinear partial differential equationsrdquoPhysics Letters A vol 286 no 2-3 pp 175ndash179 2001

[8] R Hirota The Direct Method in Soliton Theory vol 155 ofCambridge Tracts in Mathematics Cambridge University PressCambridge UK 2004

[9] M Wang X Li and J Zhang ldquoThe (1198661015840

119866)-expansion methodand travelling wave solutions of nonlinear evolution equationsin mathematical physicsrdquo Physics Letters A vol 372 no 4 pp417ndash423 2008

[10] D Lu ldquoJacobi elliptic function solutions for two variant Boussi-nesq equationsrdquo Chaos Solitons amp Fractals vol 24 no 5 pp1373ndash1385 2005

[11] Z Yan ldquoAbundant families of Jacobi elliptic function solutionsof the (2 + 1)-dimensional integrable Davey-Stewartson-typeequation via a new methodrdquo Chaos Solitons and Fractals vol18 no 2 pp 299ndash309 2003

[12] S-Y Lou and J Lu ldquoSpecial solutions from the variableseparation approach the Davey-Stewartson equationrdquo Journalof Physics A Mathematical and General vol 29 no 14 pp4209ndash4215 1996

[13] A-M Wazwaz ldquoThe tanh and the sine-cosine methods forcompact and noncompact solutions of the nonlinear Klein-Gordon equationrdquo Applied Mathematics and Computation vol167 no 2 pp 1179ndash1195 2005

[14] Z Yan ldquoThe new tri-function method to multiple exact solu-tions of nonlinear wave equationsrdquo Physica Scripta vol 78 no3 Article ID 035001 5 pages 2008

[15] Z Yan ldquoPeriodic solitary and rational wave solutions of the3D extended quantum Zakharov-Kuznetsov equation in densequantum plasmasrdquo Physics Letters Section A General Atomicand Solid State Physics vol 373 no 29 pp 2432ndash2437 2009

[16] M Wang and X Li ldquoExtended 119865-expansion method and peri-odic wave solutions for the generalized Zakharov equationsrdquoPhysics Letters A vol 343 no 1ndash3 pp 48ndash54 2005

[17] S Zhang ldquoApplication of Exp-function method to high-dimensional nonlinear evolution equationrdquo Chaos Solitons andFractals vol 38 no 1 pp 270ndash276 2008

[18] W-X Ma T Huang and Y Zhang ldquoA multiple exp-functionmethod for nonlinear differential equations and its applicationrdquoPhysica Scripta vol 82 no 6 Article ID 065003 2010

[19] G W Bluman and S Kumei Symmetries and DifferentialEquations AppliedMathematical Sciences Springer NewYorkNY USA 1989

[20] P J Olver Applications of Lie Groups to Differential Equationsvol 107 of Graduate Texts in Mathematics Springer New YorkNY USA 2nd edition 1993

[21] L V Ovsiannikov Group Analysis of Differential EquationsAcademic Press New York NY USA 1982

[22] N H Ibragimov CRC Handbook of Lie Group Analysis ofDifferential Equations vol 1 CRC Press Boca Raton Fla USA1994

[23] N H Ibragimov CRC Handbook of Lie Group Analysis ofDifferential Equations vol 2 CRC Press Boca Raton Fla USA1995

[24] N H Ibragimov CRC Handbook of Lie Group Analysis ofDifferential Equations vol 3 CRC Press Boca Raton Fla USA1996

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

[26] A-M Wazwaz ldquoThe tanh method and the sine-cosine methodfor solving the KP-MEW equationrdquo International Journal ofComputer Mathematics vol 82 no 2 pp 235ndash246 2005

[27] A Saha ldquoBifurcation of travelling wave solutions for the gener-alized KP-NEW equationsrdquo Communications in Nonlinear Sci-ence and Numerical Simulation vol 17 no 9 pp 3539ndash35512012

[28] MWei S Tang H Fu and G Chen ldquoSingle peak solitary wavesolutions for the generalized KP-MEW (2 2) equation under

Mathematical Problems in Engineering 7

boundary conditionrdquo Applied Mathematics and Computationvol 219 no 17 pp 8979ndash8990 2013

[29] S C Anco and G Bluman ldquoDirect construction method forconservation laws of partial differential equations Part I exam-ples of conservation law classificationsrdquo European Journal ofApplied Mathematics vol 13 no 5 pp 545ndash566 2002

[30] K R Adem and C M Khalique ldquoConservation laws andtraveling wave solutions of a generalized nonlinear ZK-BBMequationrdquo Abstract and Applied Analysis vol 2014 Article ID139513 5 pages 2014

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

Mathematical Problems in Engineering 7

boundary conditionrdquo Applied Mathematics and Computationvol 219 no 17 pp 8979ndash8990 2013

[29] S C Anco and G Bluman ldquoDirect construction method forconservation laws of partial differential equations Part I exam-ples of conservation law classificationsrdquo European Journal ofApplied Mathematics vol 13 no 5 pp 545ndash566 2002

[30] K R Adem and C M Khalique ldquoConservation laws andtraveling wave solutions of a generalized nonlinear ZK-BBMequationrdquo Abstract and Applied Analysis vol 2014 Article ID139513 5 pages 2014

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