Hindawi Publishing CorporationISRNMathematical PhysicsVolume 2013 Article ID 146704 5 pageshttpdxdoiorg1011552013146704
Research ArticleThe Modified Simple Equation Method for Exact andSolitary Wave Solutions of Nonlinear Evolution EquationThe GZK-BBM Equation and Right-Handed NoncommutativeBurgers Equations
Kamruzzaman Khan1 M Ali Akbar23 and Norhashidah Hj Mohd Ali3
1 Department of Mathematics Pabna Science and Technology University Pabna 6600 Bangladesh2Department of Applied Mathematics University of Rajshahi Rajshahi 6205 Bangladesh3 School of Mathematical Sciences Universiti Sains Malaysia 11800 USM Pulau Pinang Malaysia
Correspondence should be addressed to M Ali Akbar ali math74yahoocom
Received 25 November 2012 Accepted 10 January 2013
Academic Editors A Herrera-Aguilar W-H Steeb and H Zhou
Copyright copy 2013 Kamruzzaman Khan et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
The modified simple equation method is significant for finding the exact traveling wave solutions of nonlinear evolution equations(NLEEs) in mathematical physics In this paper we bring in the modified simple equation (MSE) method for solving NLEEsvia the Generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony (GZK-BBM) equation and the right-handed noncommutativeBurgersrsquo (nc-Burgers) equations and achieve the exact solutions involving parameters When the parameters are taken as specialvalues the solitary wave solutions are originated from the traveling wave solutions It is established that the MSE method offers afurther influential mathematical tool for constructing the exact solutions of NLEEs in mathematical physics
1 Introduction
The importance of nonlinear evolution equations (NLEEs) isnow well established since these equations arise in variousareas of science and engineering especially in fluid mechan-ics biology plasma physics solid-state physics optical fibersbiophysics and so on As a key problem finding theiranalytical solutions is of great importance and is actuallyexecuted through various efficient and powerful methodssuch as the Exp-function method [1ndash4] the tanh-functionmethod [5 6] the homogeneous balance method [7 8]the (1198661015840119866)-expansion method [9ndash16] the Hirotarsquos bilineartransformationmethod [17 18] the Backlund transformationmethod [19] the inverse scattering transformation [20] theJacobi elliptic function method [21] the modified simpleequation method [22ndash24] and so on
Theobjective of this paper is to look for new study relatingto the MSE method via the well-recognized GZK-BBM
equation and right-handed nc-Burgersrsquo equation and estab-lish the originality and effectiveness of the method
The paper is organized as follows in Section 2 we givethe description of the MSE method In Section 3 we usethis method to the nonlinear evolution equations pointed outabove and in Section 4 conclusions are given
2 Description of the MSE Method
Suppose the nonlinear evolution equation is in the followingform
where 119865 is a polynomial of 119906(119909 119905) and its partial derivativeswherein the highest order derivatives and nonlinear terms areconcernedThemain steps of theMSEmethod [22ndash24] are asfollows
where 119875 is a polynomial in 119906(120585) and its total derivativeswherein 1199061015840(120585) = 119889119906119889120585
Step 2 We suppose the solution of (3) is of the form
119906 (120585) =
119873
sum119896=0
119862119896[Φ1015840 (120585)
Φ (120585)]
119896
(4)
where 119862119896(119896 = 0 1 2 3 ) are arbitrary constants to be
determined such that 119862119873= 0 and Φ(120585) is an unidenti-
fied function to be determined afterwards In Exp-functionmethod (1198661015840119866)-expansion method tanh-function methodJacobi elliptic function method and so forth the solutionis offered in terms of some predefined functions but inthe MSE method Φ is not predefined or not a solution ofany predefined differential equation Therefore some freshsolutions might be found by this method This is the meritof the MSE method
Step 3 We determine the positive integer 119873 come out in(4) by considering the homogeneous balance between thehighest order derivatives and the highest order nonlinearterms occurring in (3)
Step 4 We compute all the required derivatives 1199061015840 11990610158401015840 and substitute (4) and the derivatives into (3) and then weaccount for the functionΦ(120585) As a result of this substitutionwe get a polynomial of (Φ1015840(120585)Φ(120585)) and its derivatives Inthis polynomial we equate all the coefficients to zero Thisprocedure yields a system of equations whichever can besolved to find 119862
119896and Φ(120585)
3 Applications
31 The GZK-BBM Equation In this subsection we will usethe MSE method to look for the exact solutions and then thesolitary wave solutions to the GZK-BBM equation
119906119905+ 119906119909+ 120572(119906
3)119909+ 120573(119906
119909119905+ 119906119910119910)119909= 0 (5)
where 120572 and 120573 are nonzero constantsUsing traveling wave transformation
(1 minus 120584) 119906 + 1205721199063+ 120573 (1 minus 120584) 119906
10158401015840= 0 (8)
Balancing the highest order derivative 11990610158401015840 and nonlinear term1199063 we obtain119873 = 1
Therefore the solution (4) turns into the following form
119906 (120585) = 1198620+ 1198621(Φ1015840
Φ) (9)
where 1198620and 119862
1are constants such that 119862
1= 0 and Φ(120585) is
an unidentified function to be determined It is easy to findthat
1199061015840= 1198621[Φ10158401015840
Φminus (
Φ1015840
Φ)
2
] (10)
11990610158401015840= 1198621
Φ101584010158401015840
Φminus 31198621
Φ10158401015840Φ1015840
Φ2+ 21198601(Φ1015840
Φ)
3
(11)
1199063= 1198621
3(Φ1015840
Φ)
3
+ 31198621
21198620(Φ1015840
Φ)
2
+ 311986211198620
2(Φ1015840
Φ) + 119862
0
3
(12)
Substituting the values of 119906 11990610158401015840 and 1199063 from (9)ndash(12) into (8)and then equating the coefficients of Φ0 Φminus1 Φminus2 and Φminus3to zero we obtain
1205721198620
3+ 1198620minus 1205841198620= 0 (13)
120573 (120584 minus 1)Φ101584010158401015840minus (3120572119862
0
2minus 120584 + 1)Φ
1015840= 0 (14)
120573 (120584 minus 1)Φ10158401015840+ 12057211986201198621Φ1015840= 0 (15)
(1205721198621
3+ 2120573119862
1minus 2120573120584119862
1) (Φ1015840)3
= 0 (16)
Equations (13) and (16) respectively yield
1198620= 0 plusmnradic
120584 minus 1
120572 119862
1= plusmnradic
2120573 (120584 minus 1)
120572since 119862
1= 0
(17)
where 120584 = 1From (14) and (15) we obtain
Φ101584010158401015840
Φ10158401015840= minus119897 (18)
where 119897 = (31205721198620
2 minus 120584 + 1)12057211986201198621
Integrating (18) we obtain
Φ10158401015840
(120585) = 1198881119890minus119897120585 (19)
where 1198881is a constant of integration
And from (15) and (19) we obtain
Φ1015840= minus119898119890
minus119897120585 (20)
where119898 = 120573(120584 minus 1)119888112057211986201198621
ISRNMathematical Physics 3
Integrating (20) with respect to 120585 we obtain
Φ (120585) = 1198882+119898
119897119890minus119897120585 (21)
where 1198882is a constant of integration
Substituting the value ofΦ andΦ1015840 into solution (9) yields
For 119910 = 0 the solution 119906(119909 119910 119905) presented in (26) issketched in Figure 1
Again for 119910 = 0 the solution 119906(119909 119910 119905) presented in (27)is sketched in Figure 2
The MSE method is applied to investigate solitary wavesolutions to the GZK-BBM equation and obtained solutionswith free parameters involving the known solutions in theopen literature Obviously we might choose the values of thearbitrary constants 119888
1and 1198882equal to other values resulting
in diverse solitary shapes The free parameters imply somephysical meaningful results in gravity water waves in thelong-wave regime
2
15
1
050
minus05
minus1
minus15
minus2
2
32
1
1
0
0
minus1
minus1
minus2
minus2
minus3minus3
119906
119905
119909
Figure 1
15
10
5
0
minus5
minus10
minus15
240
minus2
24
0minus2
minus4minus4
119906
119905
119909
Figure 2
32 The Right-Handed nc-Burgersrsquo Equation In this subsec-tion we will bring to bear the MSE method to find thetraveling wave solutions and then the solitary wave solutionsto the right-handed nc-Burgersrsquo equation
119906119905= 119906119909119909+ 2119906119906
119909 (28)
Using traveling wave transformation (2) (28) is reduced tothe following ODE
11990610158401015840+ 2119906119906
1015840+ 1205841199061015840= 0 (29)
Integrating (29) with respect to 120585 and setting the constant ofintegration to zero we obtain
1199061015840+ 1199062+ 120584119906 = 0 (30)
Balancing the highest order derivative and nonlinear termwe obtain119873 = 1
Therefore solution (4) becomes
119906 (120585) = 1198620+ 1198621(Φ1015840
Φ) (31)
4 ISRNMathematical Physics
Executing the parallel course of action which is described inSection 31 we obtain
1205841198620+ 1198620
2= 0 (32)
Φ10158401015840+ (2119862
0+ 120584)Φ
1015840= 0 (33)
1198621
2minus 1198621= 0 (34)
Solving (32) and (34) we obtain 1198620= 0 minus120584 and 119862
1= 1
since 1198621= 0 respectively
Case 1 When 1198620= 0 and 119862
1= 1 and solving (33) we receive
the value of Φ and substituting the value of Φ into (31) weobtain the following exact solution
119906 (119909 119905) =120584 1198881exp (minus120584 (119909 minus 120584 119905))
120584 1198882minus 1198881exp (minus120584 (119909 minus 120584 119905))
(35)
where 1198881and 1198882are constants of integrationTherefore we can
make choices at random the parameters 1198881and 1198882 if we choose
1198881= 120584 and 119888
2= 1 the exact solution (35) turns into the under
determined solitary wave solution
119906 (119909 119905) =minus120584
21 minus coth(120584(119909 minus 120584119905)
2) (36)
And if 1198881= minus120584 and 119888
2= 1 the solution (35) turn into
119906 (119909 119905) =minus120584
21 minus tanh(120584(119909 minus 120584119905)
2) (37)
Case 2 When1198620= minus120584 and solving (33) we get the value ofΦ
and substituting this value into (31) we obtain the subsequentexact solution
119906 (119909 119905) = minus120584 +1205841198881exp (120584 (119909 minus 120584119905))
1198881exp (120584 (119909 minus 120584119905)) + 120584119888
2
(38)
We can arbitrarily pick the parameters 1198881and 1198882 Therefore
exact solution (38) turns into the following solitary wavesolutions
119906 (119909 119905) = minus120584 +120584
21 + tanh(120584(119909 minus 120584119905)
2) (39)
when 1198881= 120584 and 119888
2= 1 and
119906 (119909 119905) = minus120584 +120584
21 + coth(120584 (119909 minus 120584119905)
2) (40)
when 1198881= minus120584 and 119888
2= 1
The solution 119906(119909 119905) given in (39) is presented in Figure 3The solution 119906(119909 119905) given in (40) is presented in Figure 4
For specific values of the parameters in the generalizedexact solutions (35) and (38) we obtain the solitary waveshape solutions to the right-handed nc-Burgersrsquo equationwhich are shown in Figures 3 and 4 Of course we mightchoose other values of the arbitrary constants 119888
1and 1198882 result-
ing in diverse solitary wave shapes The free parameters mayimply some physical meaningful results in fluid mechanicsgas dynamics and traffic flow
0
minus05
10
5
5
0
0minus10
minus10
minus5
minus5
minus1
minus15
minus2
minus25
minus3
minus35
119906
119905
119909
Figure 3
1055 00minus10minus10
minus5minus5
119906
119905119909
15119890 + 15
1119890 + 15
5119890 + 14
0
minus5119890 + 14
minus1119890 + 15
minus15119890 + 15
Figure 4
4 Conclusions
Themodified simple equationmethod presented in this paperhas been successfully implemented to find the exact and thesolitary wave solutions for NLEEs via the GZK-BBM andright-handed nc-Burgersrsquo equation The method offers solu-tions with free parameters that might be important to explainsome intricate physical phenomena Some special solutionsincluding the known solitary wave solution are originatedby setting appropriate values for the parameters Comparedto the currently proposed method with other methods suchas the (1198661015840119866)-expansion method the Exp-function methodand the tanh-function method we might conclude that theexact solutions to (5) and (28) can be investigated usingthese methods with the help of the symbolic computationalsoftware such as Mathematica and Maple to facilitate thecomplex algebraic computations On the other hand viathe proposed method the exact and solitary wave solutionsto these equations have been achieved without using any
ISRNMathematical Physics 5
symbolic computation software because the method is verysimple and has easy computations
References
[1] J H He and X H Wu ldquoExp-function method for nonlinearwave equationsrdquo Chaos Solitons amp Fractals vol 30 no 3 pp700ndash708 2006
[2] M A Akbar and N H M Ali ldquoExp-function method for duff-ing equation and new solutions of (2+1) dimensional dispersivelong Wave Equationsrdquo Progress in Applied Mathematics vol 1no 2 pp 30ndash42 2011
[3] H Naher F A Abdullah and M Ali Akbar ldquoNew travelingwave solutions of the higher dimensional nonlinear partialdifferential equation by the exp-function methodrdquo Journal ofApplied Mathematics Article ID 575387 14 pages 2012
[4] H Naher A F Abdullah and M A Akbar ldquoThe Exp-functionmethod for new exact solutions of the nonlinear partial differ-ential equationsrdquo International Journal of the Physical Sciencesvol 6 no 29 pp 6706ndash6716 2011
[5] M A Abdou ldquoThe extended tanh method and its applicationsfor solving nonlinear physical modelsrdquo Applied Mathematicsand Computation vol 190 no 1 pp 988ndash996 2007
[6] E Fan ldquoExtended tanh-function method and its applicationsto nonlinear equationsrdquo Physics Letters A vol 277 no 4-5 pp212ndash218 2000
[7] M L Wang ldquoSolitary wave solutions for variant Boussinesqequationsrdquo Physics Letters A vol 199 no 3-4 pp 169ndash172 1995
[8] EM E Zayed H A Zedan and K A Gepreel ldquoOn the solitarywave solutions for nonlinear Hirota-Satsuma coupled KdV ofequationsrdquo Chaos Solitons amp Fractals vol 22 no 2 pp 285ndash303 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] E M E Zayed and K A Gepreel ldquoThe (1198661015840
119866)-expansionmethod for finding traveling wave solutions of nonlinearpartial differential equations in mathematical physicsrdquo Journalof Mathematical Physics vol 50 no 1 pp 013502ndash013514 2009
[11] E M E Zayed ldquoTraveling wave solutions for higherdimensional nonlinear evolution equations using the (119866
[14] M A Akbar N HM Ali and S T Mohyud-Din ldquoThe alterna-tive (119866
1015840
119866)-expansion method with generalized Riccati equa-tion application to fifth order (1 + 1)-dimensional Caudrey-Dodd-Gibbon equationrdquo International Journal of Physical Sci-ences vol 7 no 5 pp 743ndash752 2012
[15] M A Akbar and N H M Ali ldquoThe alternative (1198661015840
119866)-expansion method and its applications to nonlinear partial
differential equationsrdquo International Journal of Physical Sciencesvol 6 no 35 pp 7910ndash7920 2011
[16] M A Akbar N H M Ali and S T Mohyud-Din ldquoSomenew exact traveling wave solutions to the (3 + 1)-dimensionalKadomtsev-Petviashvili equationrdquoWorld Applied Sciences Jour-nal vol 16 no 11 pp 1551ndash1558 2012
[17] RHirota ldquoExact envelope-soliton solutions of a nonlinear waveequationrdquo Journal of Mathematical Physics vol 14 pp 805ndash8091973
[18] R Hirota and J Satsuma ldquoSoliton solutions of a coupledKorteweg-de Vries equationrdquo Physics Letters A vol 85 no 8-9pp 407ndash408 1981
[19] M R Miura Backlund Transformation Springer Berlin Ger-many 1978
[20] M J Ablowitz and P A Clarkson Solitons nonlinear evolutionequations and inverse scattering vol 149 of London Mathemat-ical Society Lecture Note Series Cambridge University PressCambridge UK 1991
[21] D Lu and Q Shi ldquoNew Jacobi elliptic functions solutions forthe combined KdV-MKdV equationrdquo International Journal ofNonlinear Science vol 10 no 3 pp 320ndash325 2010
[22] A J M Jawad M D Petkovic and A Biswas ldquoModified simpleequation method for nonlinear evolution equationsrdquo AppliedMathematics and Computation vol 217 no 2 pp 869ndash877 2010
[23] EM E Zayed ldquoAnote on themodified simple equationmethodapplied to Sharma-Tasso-Olver equationrdquo Applied Mathematicsand Computation vol 218 no 7 pp 3962ndash3964 2011
[24] E M E Zayed and S A H Ibrahim ldquoExact solutions ofnonlinear evolution equations in mathematical physics usingthe modified simple equation methodrdquo Chinese Physics Lettersvol 29 no 6 Article ID 060201 2012
where 119875 is a polynomial in 119906(120585) and its total derivativeswherein 1199061015840(120585) = 119889119906119889120585
Step 2 We suppose the solution of (3) is of the form
119906 (120585) =
119873
sum119896=0
119862119896[Φ1015840 (120585)
Φ (120585)]
119896
(4)
where 119862119896(119896 = 0 1 2 3 ) are arbitrary constants to be
determined such that 119862119873= 0 and Φ(120585) is an unidenti-
fied function to be determined afterwards In Exp-functionmethod (1198661015840119866)-expansion method tanh-function methodJacobi elliptic function method and so forth the solutionis offered in terms of some predefined functions but inthe MSE method Φ is not predefined or not a solution ofany predefined differential equation Therefore some freshsolutions might be found by this method This is the meritof the MSE method
Step 3 We determine the positive integer 119873 come out in(4) by considering the homogeneous balance between thehighest order derivatives and the highest order nonlinearterms occurring in (3)
Step 4 We compute all the required derivatives 1199061015840 11990610158401015840 and substitute (4) and the derivatives into (3) and then weaccount for the functionΦ(120585) As a result of this substitutionwe get a polynomial of (Φ1015840(120585)Φ(120585)) and its derivatives Inthis polynomial we equate all the coefficients to zero Thisprocedure yields a system of equations whichever can besolved to find 119862
119896and Φ(120585)
3 Applications
31 The GZK-BBM Equation In this subsection we will usethe MSE method to look for the exact solutions and then thesolitary wave solutions to the GZK-BBM equation
119906119905+ 119906119909+ 120572(119906
3)119909+ 120573(119906
119909119905+ 119906119910119910)119909= 0 (5)
where 120572 and 120573 are nonzero constantsUsing traveling wave transformation
(1 minus 120584) 119906 + 1205721199063+ 120573 (1 minus 120584) 119906
10158401015840= 0 (8)
Balancing the highest order derivative 11990610158401015840 and nonlinear term1199063 we obtain119873 = 1
Therefore the solution (4) turns into the following form
119906 (120585) = 1198620+ 1198621(Φ1015840
Φ) (9)
where 1198620and 119862
1are constants such that 119862
1= 0 and Φ(120585) is
an unidentified function to be determined It is easy to findthat
1199061015840= 1198621[Φ10158401015840
Φminus (
Φ1015840
Φ)
2
] (10)
11990610158401015840= 1198621
Φ101584010158401015840
Φminus 31198621
Φ10158401015840Φ1015840
Φ2+ 21198601(Φ1015840
Φ)
3
(11)
1199063= 1198621
3(Φ1015840
Φ)
3
+ 31198621
21198620(Φ1015840
Φ)
2
+ 311986211198620
2(Φ1015840
Φ) + 119862
0
3
(12)
Substituting the values of 119906 11990610158401015840 and 1199063 from (9)ndash(12) into (8)and then equating the coefficients of Φ0 Φminus1 Φminus2 and Φminus3to zero we obtain
1205721198620
3+ 1198620minus 1205841198620= 0 (13)
120573 (120584 minus 1)Φ101584010158401015840minus (3120572119862
0
2minus 120584 + 1)Φ
1015840= 0 (14)
120573 (120584 minus 1)Φ10158401015840+ 12057211986201198621Φ1015840= 0 (15)
(1205721198621
3+ 2120573119862
1minus 2120573120584119862
1) (Φ1015840)3
= 0 (16)
Equations (13) and (16) respectively yield
1198620= 0 plusmnradic
120584 minus 1
120572 119862
1= plusmnradic
2120573 (120584 minus 1)
120572since 119862
1= 0
(17)
where 120584 = 1From (14) and (15) we obtain
Φ101584010158401015840
Φ10158401015840= minus119897 (18)
where 119897 = (31205721198620
2 minus 120584 + 1)12057211986201198621
Integrating (18) we obtain
Φ10158401015840
(120585) = 1198881119890minus119897120585 (19)
where 1198881is a constant of integration
And from (15) and (19) we obtain
Φ1015840= minus119898119890
minus119897120585 (20)
where119898 = 120573(120584 minus 1)119888112057211986201198621
ISRNMathematical Physics 3
Integrating (20) with respect to 120585 we obtain
Φ (120585) = 1198882+119898
119897119890minus119897120585 (21)
where 1198882is a constant of integration
Substituting the value ofΦ andΦ1015840 into solution (9) yields
For 119910 = 0 the solution 119906(119909 119910 119905) presented in (26) issketched in Figure 1
Again for 119910 = 0 the solution 119906(119909 119910 119905) presented in (27)is sketched in Figure 2
The MSE method is applied to investigate solitary wavesolutions to the GZK-BBM equation and obtained solutionswith free parameters involving the known solutions in theopen literature Obviously we might choose the values of thearbitrary constants 119888
1and 1198882equal to other values resulting
in diverse solitary shapes The free parameters imply somephysical meaningful results in gravity water waves in thelong-wave regime
2
15
1
050
minus05
minus1
minus15
minus2
2
32
1
1
0
0
minus1
minus1
minus2
minus2
minus3minus3
119906
119905
119909
Figure 1
15
10
5
0
minus5
minus10
minus15
240
minus2
24
0minus2
minus4minus4
119906
119905
119909
Figure 2
32 The Right-Handed nc-Burgersrsquo Equation In this subsec-tion we will bring to bear the MSE method to find thetraveling wave solutions and then the solitary wave solutionsto the right-handed nc-Burgersrsquo equation
119906119905= 119906119909119909+ 2119906119906
119909 (28)
Using traveling wave transformation (2) (28) is reduced tothe following ODE
11990610158401015840+ 2119906119906
1015840+ 1205841199061015840= 0 (29)
Integrating (29) with respect to 120585 and setting the constant ofintegration to zero we obtain
1199061015840+ 1199062+ 120584119906 = 0 (30)
Balancing the highest order derivative and nonlinear termwe obtain119873 = 1
Therefore solution (4) becomes
119906 (120585) = 1198620+ 1198621(Φ1015840
Φ) (31)
4 ISRNMathematical Physics
Executing the parallel course of action which is described inSection 31 we obtain
1205841198620+ 1198620
2= 0 (32)
Φ10158401015840+ (2119862
0+ 120584)Φ
1015840= 0 (33)
1198621
2minus 1198621= 0 (34)
Solving (32) and (34) we obtain 1198620= 0 minus120584 and 119862
1= 1
since 1198621= 0 respectively
Case 1 When 1198620= 0 and 119862
1= 1 and solving (33) we receive
the value of Φ and substituting the value of Φ into (31) weobtain the following exact solution
119906 (119909 119905) =120584 1198881exp (minus120584 (119909 minus 120584 119905))
120584 1198882minus 1198881exp (minus120584 (119909 minus 120584 119905))
(35)
where 1198881and 1198882are constants of integrationTherefore we can
make choices at random the parameters 1198881and 1198882 if we choose
1198881= 120584 and 119888
2= 1 the exact solution (35) turns into the under
determined solitary wave solution
119906 (119909 119905) =minus120584
21 minus coth(120584(119909 minus 120584119905)
2) (36)
And if 1198881= minus120584 and 119888
2= 1 the solution (35) turn into
119906 (119909 119905) =minus120584
21 minus tanh(120584(119909 minus 120584119905)
2) (37)
Case 2 When1198620= minus120584 and solving (33) we get the value ofΦ
and substituting this value into (31) we obtain the subsequentexact solution
119906 (119909 119905) = minus120584 +1205841198881exp (120584 (119909 minus 120584119905))
1198881exp (120584 (119909 minus 120584119905)) + 120584119888
2
(38)
We can arbitrarily pick the parameters 1198881and 1198882 Therefore
exact solution (38) turns into the following solitary wavesolutions
119906 (119909 119905) = minus120584 +120584
21 + tanh(120584(119909 minus 120584119905)
2) (39)
when 1198881= 120584 and 119888
2= 1 and
119906 (119909 119905) = minus120584 +120584
21 + coth(120584 (119909 minus 120584119905)
2) (40)
when 1198881= minus120584 and 119888
2= 1
The solution 119906(119909 119905) given in (39) is presented in Figure 3The solution 119906(119909 119905) given in (40) is presented in Figure 4
For specific values of the parameters in the generalizedexact solutions (35) and (38) we obtain the solitary waveshape solutions to the right-handed nc-Burgersrsquo equationwhich are shown in Figures 3 and 4 Of course we mightchoose other values of the arbitrary constants 119888
1and 1198882 result-
ing in diverse solitary wave shapes The free parameters mayimply some physical meaningful results in fluid mechanicsgas dynamics and traffic flow
0
minus05
10
5
5
0
0minus10
minus10
minus5
minus5
minus1
minus15
minus2
minus25
minus3
minus35
119906
119905
119909
Figure 3
1055 00minus10minus10
minus5minus5
119906
119905119909
15119890 + 15
1119890 + 15
5119890 + 14
0
minus5119890 + 14
minus1119890 + 15
minus15119890 + 15
Figure 4
4 Conclusions
Themodified simple equationmethod presented in this paperhas been successfully implemented to find the exact and thesolitary wave solutions for NLEEs via the GZK-BBM andright-handed nc-Burgersrsquo equation The method offers solu-tions with free parameters that might be important to explainsome intricate physical phenomena Some special solutionsincluding the known solitary wave solution are originatedby setting appropriate values for the parameters Comparedto the currently proposed method with other methods suchas the (1198661015840119866)-expansion method the Exp-function methodand the tanh-function method we might conclude that theexact solutions to (5) and (28) can be investigated usingthese methods with the help of the symbolic computationalsoftware such as Mathematica and Maple to facilitate thecomplex algebraic computations On the other hand viathe proposed method the exact and solitary wave solutionsto these equations have been achieved without using any
ISRNMathematical Physics 5
symbolic computation software because the method is verysimple and has easy computations
References
[1] J H He and X H Wu ldquoExp-function method for nonlinearwave equationsrdquo Chaos Solitons amp Fractals vol 30 no 3 pp700ndash708 2006
[2] M A Akbar and N H M Ali ldquoExp-function method for duff-ing equation and new solutions of (2+1) dimensional dispersivelong Wave Equationsrdquo Progress in Applied Mathematics vol 1no 2 pp 30ndash42 2011
[3] H Naher F A Abdullah and M Ali Akbar ldquoNew travelingwave solutions of the higher dimensional nonlinear partialdifferential equation by the exp-function methodrdquo Journal ofApplied Mathematics Article ID 575387 14 pages 2012
[4] H Naher A F Abdullah and M A Akbar ldquoThe Exp-functionmethod for new exact solutions of the nonlinear partial differ-ential equationsrdquo International Journal of the Physical Sciencesvol 6 no 29 pp 6706ndash6716 2011
[5] M A Abdou ldquoThe extended tanh method and its applicationsfor solving nonlinear physical modelsrdquo Applied Mathematicsand Computation vol 190 no 1 pp 988ndash996 2007
[6] E Fan ldquoExtended tanh-function method and its applicationsto nonlinear equationsrdquo Physics Letters A vol 277 no 4-5 pp212ndash218 2000
[7] M L Wang ldquoSolitary wave solutions for variant Boussinesqequationsrdquo Physics Letters A vol 199 no 3-4 pp 169ndash172 1995
[8] EM E Zayed H A Zedan and K A Gepreel ldquoOn the solitarywave solutions for nonlinear Hirota-Satsuma coupled KdV ofequationsrdquo Chaos Solitons amp Fractals vol 22 no 2 pp 285ndash303 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] E M E Zayed and K A Gepreel ldquoThe (1198661015840
119866)-expansionmethod for finding traveling wave solutions of nonlinearpartial differential equations in mathematical physicsrdquo Journalof Mathematical Physics vol 50 no 1 pp 013502ndash013514 2009
[11] E M E Zayed ldquoTraveling wave solutions for higherdimensional nonlinear evolution equations using the (119866
[14] M A Akbar N HM Ali and S T Mohyud-Din ldquoThe alterna-tive (119866
1015840
119866)-expansion method with generalized Riccati equa-tion application to fifth order (1 + 1)-dimensional Caudrey-Dodd-Gibbon equationrdquo International Journal of Physical Sci-ences vol 7 no 5 pp 743ndash752 2012
[15] M A Akbar and N H M Ali ldquoThe alternative (1198661015840
119866)-expansion method and its applications to nonlinear partial
differential equationsrdquo International Journal of Physical Sciencesvol 6 no 35 pp 7910ndash7920 2011
[16] M A Akbar N H M Ali and S T Mohyud-Din ldquoSomenew exact traveling wave solutions to the (3 + 1)-dimensionalKadomtsev-Petviashvili equationrdquoWorld Applied Sciences Jour-nal vol 16 no 11 pp 1551ndash1558 2012
[17] RHirota ldquoExact envelope-soliton solutions of a nonlinear waveequationrdquo Journal of Mathematical Physics vol 14 pp 805ndash8091973
[18] R Hirota and J Satsuma ldquoSoliton solutions of a coupledKorteweg-de Vries equationrdquo Physics Letters A vol 85 no 8-9pp 407ndash408 1981
[19] M R Miura Backlund Transformation Springer Berlin Ger-many 1978
[20] M J Ablowitz and P A Clarkson Solitons nonlinear evolutionequations and inverse scattering vol 149 of London Mathemat-ical Society Lecture Note Series Cambridge University PressCambridge UK 1991
[21] D Lu and Q Shi ldquoNew Jacobi elliptic functions solutions forthe combined KdV-MKdV equationrdquo International Journal ofNonlinear Science vol 10 no 3 pp 320ndash325 2010
[22] A J M Jawad M D Petkovic and A Biswas ldquoModified simpleequation method for nonlinear evolution equationsrdquo AppliedMathematics and Computation vol 217 no 2 pp 869ndash877 2010
[23] EM E Zayed ldquoAnote on themodified simple equationmethodapplied to Sharma-Tasso-Olver equationrdquo Applied Mathematicsand Computation vol 218 no 7 pp 3962ndash3964 2011
[24] E M E Zayed and S A H Ibrahim ldquoExact solutions ofnonlinear evolution equations in mathematical physics usingthe modified simple equation methodrdquo Chinese Physics Lettersvol 29 no 6 Article ID 060201 2012
For 119910 = 0 the solution 119906(119909 119910 119905) presented in (26) issketched in Figure 1
Again for 119910 = 0 the solution 119906(119909 119910 119905) presented in (27)is sketched in Figure 2
The MSE method is applied to investigate solitary wavesolutions to the GZK-BBM equation and obtained solutionswith free parameters involving the known solutions in theopen literature Obviously we might choose the values of thearbitrary constants 119888
1and 1198882equal to other values resulting
in diverse solitary shapes The free parameters imply somephysical meaningful results in gravity water waves in thelong-wave regime
2
15
1
050
minus05
minus1
minus15
minus2
2
32
1
1
0
0
minus1
minus1
minus2
minus2
minus3minus3
119906
119905
119909
Figure 1
15
10
5
0
minus5
minus10
minus15
240
minus2
24
0minus2
minus4minus4
119906
119905
119909
Figure 2
32 The Right-Handed nc-Burgersrsquo Equation In this subsec-tion we will bring to bear the MSE method to find thetraveling wave solutions and then the solitary wave solutionsto the right-handed nc-Burgersrsquo equation
119906119905= 119906119909119909+ 2119906119906
119909 (28)
Using traveling wave transformation (2) (28) is reduced tothe following ODE
11990610158401015840+ 2119906119906
1015840+ 1205841199061015840= 0 (29)
Integrating (29) with respect to 120585 and setting the constant ofintegration to zero we obtain
1199061015840+ 1199062+ 120584119906 = 0 (30)
Balancing the highest order derivative and nonlinear termwe obtain119873 = 1
Therefore solution (4) becomes
119906 (120585) = 1198620+ 1198621(Φ1015840
Φ) (31)
4 ISRNMathematical Physics
Executing the parallel course of action which is described inSection 31 we obtain
1205841198620+ 1198620
2= 0 (32)
Φ10158401015840+ (2119862
0+ 120584)Φ
1015840= 0 (33)
1198621
2minus 1198621= 0 (34)
Solving (32) and (34) we obtain 1198620= 0 minus120584 and 119862
1= 1
since 1198621= 0 respectively
Case 1 When 1198620= 0 and 119862
1= 1 and solving (33) we receive
the value of Φ and substituting the value of Φ into (31) weobtain the following exact solution
119906 (119909 119905) =120584 1198881exp (minus120584 (119909 minus 120584 119905))
120584 1198882minus 1198881exp (minus120584 (119909 minus 120584 119905))
(35)
where 1198881and 1198882are constants of integrationTherefore we can
make choices at random the parameters 1198881and 1198882 if we choose
1198881= 120584 and 119888
2= 1 the exact solution (35) turns into the under
determined solitary wave solution
119906 (119909 119905) =minus120584
21 minus coth(120584(119909 minus 120584119905)
2) (36)
And if 1198881= minus120584 and 119888
2= 1 the solution (35) turn into
119906 (119909 119905) =minus120584
21 minus tanh(120584(119909 minus 120584119905)
2) (37)
Case 2 When1198620= minus120584 and solving (33) we get the value ofΦ
and substituting this value into (31) we obtain the subsequentexact solution
119906 (119909 119905) = minus120584 +1205841198881exp (120584 (119909 minus 120584119905))
1198881exp (120584 (119909 minus 120584119905)) + 120584119888
2
(38)
We can arbitrarily pick the parameters 1198881and 1198882 Therefore
exact solution (38) turns into the following solitary wavesolutions
119906 (119909 119905) = minus120584 +120584
21 + tanh(120584(119909 minus 120584119905)
2) (39)
when 1198881= 120584 and 119888
2= 1 and
119906 (119909 119905) = minus120584 +120584
21 + coth(120584 (119909 minus 120584119905)
2) (40)
when 1198881= minus120584 and 119888
2= 1
The solution 119906(119909 119905) given in (39) is presented in Figure 3The solution 119906(119909 119905) given in (40) is presented in Figure 4
For specific values of the parameters in the generalizedexact solutions (35) and (38) we obtain the solitary waveshape solutions to the right-handed nc-Burgersrsquo equationwhich are shown in Figures 3 and 4 Of course we mightchoose other values of the arbitrary constants 119888
1and 1198882 result-
ing in diverse solitary wave shapes The free parameters mayimply some physical meaningful results in fluid mechanicsgas dynamics and traffic flow
0
minus05
10
5
5
0
0minus10
minus10
minus5
minus5
minus1
minus15
minus2
minus25
minus3
minus35
119906
119905
119909
Figure 3
1055 00minus10minus10
minus5minus5
119906
119905119909
15119890 + 15
1119890 + 15
5119890 + 14
0
minus5119890 + 14
minus1119890 + 15
minus15119890 + 15
Figure 4
4 Conclusions
Themodified simple equationmethod presented in this paperhas been successfully implemented to find the exact and thesolitary wave solutions for NLEEs via the GZK-BBM andright-handed nc-Burgersrsquo equation The method offers solu-tions with free parameters that might be important to explainsome intricate physical phenomena Some special solutionsincluding the known solitary wave solution are originatedby setting appropriate values for the parameters Comparedto the currently proposed method with other methods suchas the (1198661015840119866)-expansion method the Exp-function methodand the tanh-function method we might conclude that theexact solutions to (5) and (28) can be investigated usingthese methods with the help of the symbolic computationalsoftware such as Mathematica and Maple to facilitate thecomplex algebraic computations On the other hand viathe proposed method the exact and solitary wave solutionsto these equations have been achieved without using any
ISRNMathematical Physics 5
symbolic computation software because the method is verysimple and has easy computations
References
[1] J H He and X H Wu ldquoExp-function method for nonlinearwave equationsrdquo Chaos Solitons amp Fractals vol 30 no 3 pp700ndash708 2006
[2] M A Akbar and N H M Ali ldquoExp-function method for duff-ing equation and new solutions of (2+1) dimensional dispersivelong Wave Equationsrdquo Progress in Applied Mathematics vol 1no 2 pp 30ndash42 2011
[3] H Naher F A Abdullah and M Ali Akbar ldquoNew travelingwave solutions of the higher dimensional nonlinear partialdifferential equation by the exp-function methodrdquo Journal ofApplied Mathematics Article ID 575387 14 pages 2012
[4] H Naher A F Abdullah and M A Akbar ldquoThe Exp-functionmethod for new exact solutions of the nonlinear partial differ-ential equationsrdquo International Journal of the Physical Sciencesvol 6 no 29 pp 6706ndash6716 2011
[5] M A Abdou ldquoThe extended tanh method and its applicationsfor solving nonlinear physical modelsrdquo Applied Mathematicsand Computation vol 190 no 1 pp 988ndash996 2007
[6] E Fan ldquoExtended tanh-function method and its applicationsto nonlinear equationsrdquo Physics Letters A vol 277 no 4-5 pp212ndash218 2000
[7] M L Wang ldquoSolitary wave solutions for variant Boussinesqequationsrdquo Physics Letters A vol 199 no 3-4 pp 169ndash172 1995
[8] EM E Zayed H A Zedan and K A Gepreel ldquoOn the solitarywave solutions for nonlinear Hirota-Satsuma coupled KdV ofequationsrdquo Chaos Solitons amp Fractals vol 22 no 2 pp 285ndash303 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] E M E Zayed and K A Gepreel ldquoThe (1198661015840
119866)-expansionmethod for finding traveling wave solutions of nonlinearpartial differential equations in mathematical physicsrdquo Journalof Mathematical Physics vol 50 no 1 pp 013502ndash013514 2009
[11] E M E Zayed ldquoTraveling wave solutions for higherdimensional nonlinear evolution equations using the (119866
[14] M A Akbar N HM Ali and S T Mohyud-Din ldquoThe alterna-tive (119866
1015840
119866)-expansion method with generalized Riccati equa-tion application to fifth order (1 + 1)-dimensional Caudrey-Dodd-Gibbon equationrdquo International Journal of Physical Sci-ences vol 7 no 5 pp 743ndash752 2012
[15] M A Akbar and N H M Ali ldquoThe alternative (1198661015840
119866)-expansion method and its applications to nonlinear partial
differential equationsrdquo International Journal of Physical Sciencesvol 6 no 35 pp 7910ndash7920 2011
[16] M A Akbar N H M Ali and S T Mohyud-Din ldquoSomenew exact traveling wave solutions to the (3 + 1)-dimensionalKadomtsev-Petviashvili equationrdquoWorld Applied Sciences Jour-nal vol 16 no 11 pp 1551ndash1558 2012
[17] RHirota ldquoExact envelope-soliton solutions of a nonlinear waveequationrdquo Journal of Mathematical Physics vol 14 pp 805ndash8091973
[18] R Hirota and J Satsuma ldquoSoliton solutions of a coupledKorteweg-de Vries equationrdquo Physics Letters A vol 85 no 8-9pp 407ndash408 1981
[19] M R Miura Backlund Transformation Springer Berlin Ger-many 1978
[20] M J Ablowitz and P A Clarkson Solitons nonlinear evolutionequations and inverse scattering vol 149 of London Mathemat-ical Society Lecture Note Series Cambridge University PressCambridge UK 1991
[21] D Lu and Q Shi ldquoNew Jacobi elliptic functions solutions forthe combined KdV-MKdV equationrdquo International Journal ofNonlinear Science vol 10 no 3 pp 320ndash325 2010
[22] A J M Jawad M D Petkovic and A Biswas ldquoModified simpleequation method for nonlinear evolution equationsrdquo AppliedMathematics and Computation vol 217 no 2 pp 869ndash877 2010
[23] EM E Zayed ldquoAnote on themodified simple equationmethodapplied to Sharma-Tasso-Olver equationrdquo Applied Mathematicsand Computation vol 218 no 7 pp 3962ndash3964 2011
[24] E M E Zayed and S A H Ibrahim ldquoExact solutions ofnonlinear evolution equations in mathematical physics usingthe modified simple equation methodrdquo Chinese Physics Lettersvol 29 no 6 Article ID 060201 2012
Executing the parallel course of action which is described inSection 31 we obtain
1205841198620+ 1198620
2= 0 (32)
Φ10158401015840+ (2119862
0+ 120584)Φ
1015840= 0 (33)
1198621
2minus 1198621= 0 (34)
Solving (32) and (34) we obtain 1198620= 0 minus120584 and 119862
1= 1
since 1198621= 0 respectively
Case 1 When 1198620= 0 and 119862
1= 1 and solving (33) we receive
the value of Φ and substituting the value of Φ into (31) weobtain the following exact solution
119906 (119909 119905) =120584 1198881exp (minus120584 (119909 minus 120584 119905))
120584 1198882minus 1198881exp (minus120584 (119909 minus 120584 119905))
(35)
where 1198881and 1198882are constants of integrationTherefore we can
make choices at random the parameters 1198881and 1198882 if we choose
1198881= 120584 and 119888
2= 1 the exact solution (35) turns into the under
determined solitary wave solution
119906 (119909 119905) =minus120584
21 minus coth(120584(119909 minus 120584119905)
2) (36)
And if 1198881= minus120584 and 119888
2= 1 the solution (35) turn into
119906 (119909 119905) =minus120584
21 minus tanh(120584(119909 minus 120584119905)
2) (37)
Case 2 When1198620= minus120584 and solving (33) we get the value ofΦ
and substituting this value into (31) we obtain the subsequentexact solution
119906 (119909 119905) = minus120584 +1205841198881exp (120584 (119909 minus 120584119905))
1198881exp (120584 (119909 minus 120584119905)) + 120584119888
2
(38)
We can arbitrarily pick the parameters 1198881and 1198882 Therefore
exact solution (38) turns into the following solitary wavesolutions
119906 (119909 119905) = minus120584 +120584
21 + tanh(120584(119909 minus 120584119905)
2) (39)
when 1198881= 120584 and 119888
2= 1 and
119906 (119909 119905) = minus120584 +120584
21 + coth(120584 (119909 minus 120584119905)
2) (40)
when 1198881= minus120584 and 119888
2= 1
The solution 119906(119909 119905) given in (39) is presented in Figure 3The solution 119906(119909 119905) given in (40) is presented in Figure 4
For specific values of the parameters in the generalizedexact solutions (35) and (38) we obtain the solitary waveshape solutions to the right-handed nc-Burgersrsquo equationwhich are shown in Figures 3 and 4 Of course we mightchoose other values of the arbitrary constants 119888
1and 1198882 result-
ing in diverse solitary wave shapes The free parameters mayimply some physical meaningful results in fluid mechanicsgas dynamics and traffic flow
0
minus05
10
5
5
0
0minus10
minus10
minus5
minus5
minus1
minus15
minus2
minus25
minus3
minus35
119906
119905
119909
Figure 3
1055 00minus10minus10
minus5minus5
119906
119905119909
15119890 + 15
1119890 + 15
5119890 + 14
0
minus5119890 + 14
minus1119890 + 15
minus15119890 + 15
Figure 4
4 Conclusions
Themodified simple equationmethod presented in this paperhas been successfully implemented to find the exact and thesolitary wave solutions for NLEEs via the GZK-BBM andright-handed nc-Burgersrsquo equation The method offers solu-tions with free parameters that might be important to explainsome intricate physical phenomena Some special solutionsincluding the known solitary wave solution are originatedby setting appropriate values for the parameters Comparedto the currently proposed method with other methods suchas the (1198661015840119866)-expansion method the Exp-function methodand the tanh-function method we might conclude that theexact solutions to (5) and (28) can be investigated usingthese methods with the help of the symbolic computationalsoftware such as Mathematica and Maple to facilitate thecomplex algebraic computations On the other hand viathe proposed method the exact and solitary wave solutionsto these equations have been achieved without using any
ISRNMathematical Physics 5
symbolic computation software because the method is verysimple and has easy computations
References
[1] J H He and X H Wu ldquoExp-function method for nonlinearwave equationsrdquo Chaos Solitons amp Fractals vol 30 no 3 pp700ndash708 2006
[2] M A Akbar and N H M Ali ldquoExp-function method for duff-ing equation and new solutions of (2+1) dimensional dispersivelong Wave Equationsrdquo Progress in Applied Mathematics vol 1no 2 pp 30ndash42 2011
[3] H Naher F A Abdullah and M Ali Akbar ldquoNew travelingwave solutions of the higher dimensional nonlinear partialdifferential equation by the exp-function methodrdquo Journal ofApplied Mathematics Article ID 575387 14 pages 2012
[4] H Naher A F Abdullah and M A Akbar ldquoThe Exp-functionmethod for new exact solutions of the nonlinear partial differ-ential equationsrdquo International Journal of the Physical Sciencesvol 6 no 29 pp 6706ndash6716 2011
[5] M A Abdou ldquoThe extended tanh method and its applicationsfor solving nonlinear physical modelsrdquo Applied Mathematicsand Computation vol 190 no 1 pp 988ndash996 2007
[6] E Fan ldquoExtended tanh-function method and its applicationsto nonlinear equationsrdquo Physics Letters A vol 277 no 4-5 pp212ndash218 2000
[7] M L Wang ldquoSolitary wave solutions for variant Boussinesqequationsrdquo Physics Letters A vol 199 no 3-4 pp 169ndash172 1995
[8] EM E Zayed H A Zedan and K A Gepreel ldquoOn the solitarywave solutions for nonlinear Hirota-Satsuma coupled KdV ofequationsrdquo Chaos Solitons amp Fractals vol 22 no 2 pp 285ndash303 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] E M E Zayed and K A Gepreel ldquoThe (1198661015840
119866)-expansionmethod for finding traveling wave solutions of nonlinearpartial differential equations in mathematical physicsrdquo Journalof Mathematical Physics vol 50 no 1 pp 013502ndash013514 2009
[11] E M E Zayed ldquoTraveling wave solutions for higherdimensional nonlinear evolution equations using the (119866
[14] M A Akbar N HM Ali and S T Mohyud-Din ldquoThe alterna-tive (119866
1015840
119866)-expansion method with generalized Riccati equa-tion application to fifth order (1 + 1)-dimensional Caudrey-Dodd-Gibbon equationrdquo International Journal of Physical Sci-ences vol 7 no 5 pp 743ndash752 2012
[15] M A Akbar and N H M Ali ldquoThe alternative (1198661015840
119866)-expansion method and its applications to nonlinear partial
differential equationsrdquo International Journal of Physical Sciencesvol 6 no 35 pp 7910ndash7920 2011
[16] M A Akbar N H M Ali and S T Mohyud-Din ldquoSomenew exact traveling wave solutions to the (3 + 1)-dimensionalKadomtsev-Petviashvili equationrdquoWorld Applied Sciences Jour-nal vol 16 no 11 pp 1551ndash1558 2012
[17] RHirota ldquoExact envelope-soliton solutions of a nonlinear waveequationrdquo Journal of Mathematical Physics vol 14 pp 805ndash8091973
[18] R Hirota and J Satsuma ldquoSoliton solutions of a coupledKorteweg-de Vries equationrdquo Physics Letters A vol 85 no 8-9pp 407ndash408 1981
[19] M R Miura Backlund Transformation Springer Berlin Ger-many 1978
[20] M J Ablowitz and P A Clarkson Solitons nonlinear evolutionequations and inverse scattering vol 149 of London Mathemat-ical Society Lecture Note Series Cambridge University PressCambridge UK 1991
[21] D Lu and Q Shi ldquoNew Jacobi elliptic functions solutions forthe combined KdV-MKdV equationrdquo International Journal ofNonlinear Science vol 10 no 3 pp 320ndash325 2010
[22] A J M Jawad M D Petkovic and A Biswas ldquoModified simpleequation method for nonlinear evolution equationsrdquo AppliedMathematics and Computation vol 217 no 2 pp 869ndash877 2010
[23] EM E Zayed ldquoAnote on themodified simple equationmethodapplied to Sharma-Tasso-Olver equationrdquo Applied Mathematicsand Computation vol 218 no 7 pp 3962ndash3964 2011
[24] E M E Zayed and S A H Ibrahim ldquoExact solutions ofnonlinear evolution equations in mathematical physics usingthe modified simple equation methodrdquo Chinese Physics Lettersvol 29 no 6 Article ID 060201 2012
symbolic computation software because the method is verysimple and has easy computations
References
[1] J H He and X H Wu ldquoExp-function method for nonlinearwave equationsrdquo Chaos Solitons amp Fractals vol 30 no 3 pp700ndash708 2006
[2] M A Akbar and N H M Ali ldquoExp-function method for duff-ing equation and new solutions of (2+1) dimensional dispersivelong Wave Equationsrdquo Progress in Applied Mathematics vol 1no 2 pp 30ndash42 2011
[3] H Naher F A Abdullah and M Ali Akbar ldquoNew travelingwave solutions of the higher dimensional nonlinear partialdifferential equation by the exp-function methodrdquo Journal ofApplied Mathematics Article ID 575387 14 pages 2012
[4] H Naher A F Abdullah and M A Akbar ldquoThe Exp-functionmethod for new exact solutions of the nonlinear partial differ-ential equationsrdquo International Journal of the Physical Sciencesvol 6 no 29 pp 6706ndash6716 2011
[5] M A Abdou ldquoThe extended tanh method and its applicationsfor solving nonlinear physical modelsrdquo Applied Mathematicsand Computation vol 190 no 1 pp 988ndash996 2007
[6] E Fan ldquoExtended tanh-function method and its applicationsto nonlinear equationsrdquo Physics Letters A vol 277 no 4-5 pp212ndash218 2000
[7] M L Wang ldquoSolitary wave solutions for variant Boussinesqequationsrdquo Physics Letters A vol 199 no 3-4 pp 169ndash172 1995
[8] EM E Zayed H A Zedan and K A Gepreel ldquoOn the solitarywave solutions for nonlinear Hirota-Satsuma coupled KdV ofequationsrdquo Chaos Solitons amp Fractals vol 22 no 2 pp 285ndash303 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] E M E Zayed and K A Gepreel ldquoThe (1198661015840
119866)-expansionmethod for finding traveling wave solutions of nonlinearpartial differential equations in mathematical physicsrdquo Journalof Mathematical Physics vol 50 no 1 pp 013502ndash013514 2009
[11] E M E Zayed ldquoTraveling wave solutions for higherdimensional nonlinear evolution equations using the (119866
[14] M A Akbar N HM Ali and S T Mohyud-Din ldquoThe alterna-tive (119866
1015840
119866)-expansion method with generalized Riccati equa-tion application to fifth order (1 + 1)-dimensional Caudrey-Dodd-Gibbon equationrdquo International Journal of Physical Sci-ences vol 7 no 5 pp 743ndash752 2012
[15] M A Akbar and N H M Ali ldquoThe alternative (1198661015840
119866)-expansion method and its applications to nonlinear partial
differential equationsrdquo International Journal of Physical Sciencesvol 6 no 35 pp 7910ndash7920 2011
[16] M A Akbar N H M Ali and S T Mohyud-Din ldquoSomenew exact traveling wave solutions to the (3 + 1)-dimensionalKadomtsev-Petviashvili equationrdquoWorld Applied Sciences Jour-nal vol 16 no 11 pp 1551ndash1558 2012
[17] RHirota ldquoExact envelope-soliton solutions of a nonlinear waveequationrdquo Journal of Mathematical Physics vol 14 pp 805ndash8091973
[18] R Hirota and J Satsuma ldquoSoliton solutions of a coupledKorteweg-de Vries equationrdquo Physics Letters A vol 85 no 8-9pp 407ndash408 1981
[19] M R Miura Backlund Transformation Springer Berlin Ger-many 1978
[20] M J Ablowitz and P A Clarkson Solitons nonlinear evolutionequations and inverse scattering vol 149 of London Mathemat-ical Society Lecture Note Series Cambridge University PressCambridge UK 1991
[21] D Lu and Q Shi ldquoNew Jacobi elliptic functions solutions forthe combined KdV-MKdV equationrdquo International Journal ofNonlinear Science vol 10 no 3 pp 320ndash325 2010
[22] A J M Jawad M D Petkovic and A Biswas ldquoModified simpleequation method for nonlinear evolution equationsrdquo AppliedMathematics and Computation vol 217 no 2 pp 869ndash877 2010
[23] EM E Zayed ldquoAnote on themodified simple equationmethodapplied to Sharma-Tasso-Olver equationrdquo Applied Mathematicsand Computation vol 218 no 7 pp 3962ndash3964 2011
[24] E M E Zayed and S A H Ibrahim ldquoExact solutions ofnonlinear evolution equations in mathematical physics usingthe modified simple equation methodrdquo Chinese Physics Lettersvol 29 no 6 Article ID 060201 2012
