Hindawi Publishing CorporationInternational Journal of AnalysisVolume 2013 Article ID 723698 6 pageshttpdxdoiorg1011552013723698

Research ArticleA Note on Solitary Wave Solutions of the Nonlinear GeneralizedCamassa-Holm Equation

Lei Zhang and Xing Tao Wang

Department of Mathematics Harbin Institute of Technology Harbin 150001 China

Correspondence should be addressed to Xing Tao Wang xingtaohiteducn

Received 24 September 2012 Revised 6 January 2013 Accepted 10 January 2013

Academic Editor Baruch Cahlon

Copyright copy 2013 L Zhang and X T Wang 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

We give a simple method for applying ordinary differential equation to solve the nonlinear generalized Camassa-Holm equation119906119905+ 2119896119906

119909minus 119906119909119909119905+ 119886119906119898119906119909minus 2119906119909119906119909119909+ 119906119906119909119909119909= 0 Furthermore we give a new ansatz In the cases where 119898 = 1 2 3 the numerical

simulations demonstrate the results

1 Introduction

A new dispersive shallow water equation

119906119905+ 120581119906119909minus 119906119909119909119905+ 3119906119906

119909= 2119906119909119906119909119909+ 119906119906119909119909119909

(1)

known as the Camassa-Holm equation has been derivedby Camassa and Holm [1] They showed that the Camassa-Holm equation has peaked wave solitary solutions whichhave the first derivative discontinuity at the wave peakcalled ldquopeakonsrdquo In [2] the integral bifurcation method wasused to study a nonlinearly dispersive wave equation ofCamassa-Holm equation type In [3] attractor for a couplednonhomogeneous Camassa-Holm equation with periodicboundary condition was investigated by means of severalinequalities In [4] sufficient conditions for themodified two-component Camassa-Holm system were established In [5]a class of nonlinear fourth order analogue of a generalizedCamassa-Holm equation was studied by using sine-cosinemethod In [6] He et al studied a generalizedCamassa-Holmequation In [7] Wazwaz established solitary wave solutionsto the modified forms of Degasperis-Procesi and Camassa-Holm equations In [8] a family of Camassa-Holm equationswith distinct parameters was investigated Also many aspectsof the problems were studied by researchers [9ndash16] Authorsin [4 17] presented periodic wave solutions and travelingwave solutions for some equations In [18] Khuri investigatedthe periodic wave and peaked solitary wave solutions of thenonlinear generalized Camassa-Holm equation and gave a

ansatz for demonstrating the existence of a new class ofsolutions In [19] Tian and Song derive some new exactpeaked solitary wave solutions of the generalized Camassa-Holm equation and two types of new exact traveling wavesolutions of the generalized weakly dissipative Camassa-Holm equations In this paper we give a simple method forapplying ordinary differential equation to solve the general-ized Camassa-Holm equation and give the improved ansatzThe numerical simulation examples demonstrate that ourmethods are applicable

2 Simplification of the Nonlinear GeneralizedCamassa-Holm Equation

Consider the following nonlinear generalized Camassa-Holm equation

119906119905+ 2119896119906

119909minus 119906119909119909119905+ 119886119906119898119906119909minus 2119906119909119906119909119909+ 119906119906119909119909119909= 0 (2)

With the velocity constant 119888 we seek the traveling wavesolution of the form 119906(119909 119905) = 119907(120585) of (2) where 120585 = 119909 minus 119888119905Substituting 119906(119909 119905) = 119907(120585) into (2) we have

minus119888

d119907d120585+ 2119896

d119907d120585+ 119888

d3119907d1205853

+ 119886119907119898 d119907d120585minus 2

d119907d120585

d2119907d1205852

minus 119907

d3119907d1205853

= 0

(3)

2 International Journal of Analysis

Integrating both sides of (3) we obtain

(119907 minus 119888)

d2119907d1205852

+

1

2

(

d119907d120585)

2

=

119886

119898 + 1

119907119898+1+ (2119896 minus 119888) 119907 + 119887 (4)

where 119887 is an arbitrary constant Let d119907d120585 = 119908 Thend2119907d1205852 = 119908(d119908d119907) Substituting this into (4) yields

d119908d119907+

1

2 (119907 minus 119888)

119908 =

(119886 (119898 + 1)) 119907119898+1+ (2119896 minus 119888) 119907 + 119887

119907 minus 119888

119908minus1

(5)

Solving (4) leads to

1199082= (119907 minus 119888)

minus1119889 + int 2 [

119886

119898 + 1

119907119898+1+ (2119896 minus 119888) 119907 + 119887] d119907

(6)

where 119889 is an arbitrary constant Therefore

(

d119907d120585)

2

= (119907 minus 119888)minus1

times 119889 + [

2119886

(119898 + 1) (119898 + 2)

119907119898+2+ (2119896 minus 119888) 119907

2+ 119887119907]

(7)

From (d120585)2 = (d|120585|)2 we have

(

d119907d 10038161003816100381610038161205851003816100381610038161003816

)

2

= (119907 minus 119888)minus1

times 119889 + [

2119886

(119898 + 1) (119898 + 2)

119907119898+2+ (2119896 minus 119888) 119907

2+ 119887119907]

(8)

Therefore

intradic

119907 minus 119888

119889 + [(2119886 ((119898 + 1) (119898 + 2))) 119907119898+2+ (2119896 minus 119888) 119907

2+ 119887119907]

d119907

=

ℎ + 120585

ℎ minus 120585

ℎ +10038161003816100381610038161205851003816100381610038161003816

ℎ minus10038161003816100381610038161205851003816100381610038161003816

(9)

where ℎ is an arbitrary constantWhen 119898 = 1 (9) is the case of the Camassa-Holm

equation We take 2119896 = 119888 119886119888 minus 3119888 + 2119896 = 0 119889 = 0 and 119887 = 0then

radic3

119886

ln 119888119907

=

ℎ + 120585

ℎ minus 120585

ℎ +10038161003816100381610038161205851003816100381610038161003816

ℎ minus10038161003816100381610038161205851003816100381610038161003816

(10)

So

119907 =

119888eradic(1198863)(ℎ+120585)119888eradic(1198863)(ℎminus120585)119888eradic(1198863)(ℎ+|120585|)119888eradic(1198863)(ℎminus|120585|)

(11)

It can be checked that 119906(119909 119905) = 119888eℎ+radic(1198863)(119909minus119888119905) 119906(119909 119905) =119888eℎminusradic(1198863)(119909minus119888119905) and 119906(119909 119905) = 119888eℎ+radic(1198863)|119909minus119888119905| are infinitely greatsolutions of the Camassa-Holm equation without asymptoticbehavior [20] Only

119906 (119909 119905) = 119888eℎminusradic(1198863)|119909minus119888119905| (12)

are the solitary wave solutions of the Camassa-Holm equa-tion

When119898 = 2 we take 2119896 lt 119888 1198861198882minus6119888+12119896 = 0 119889 = 0 and119887 = 0 In the similar way in the case119898 = 1 we only choose

radic6

119886

int

1

119907radic119907 + 119888

d120593 = ℎ minus 10038161003816100381610038161205851003816100381610038161003816 (13)

then

minusradic24

119886119888

atanhradic119907 + 119888119888

= ℎ minus10038161003816100381610038161205851003816100381610038161003816 (14)

From (14) we have

119907 = 119888 tanh2 [radic 11988611988824

(10038161003816100381610038161205851003816100381610038161003816minus ℎ)] minus 119888 (15)

Therefore we have the solitary wave solutions of the general-ized Camassa-Holm equation as

119906 (119909 119905) = 119888 tanh2 [radic 11988611988824

(|119909 minus 119888119905| minus ℎ)] minus 119888 (16)

When 119898 = 3 we take 2119896 = 119888 1198861198883 minus 10119888 + 20119896 = 0 119889 = 0and 119887 = 0 then

radic10

119886

int

1

radic1199072+ 119888119907 + 119888

2

d119907 = ℎ minus 10038161003816100381610038161205851003816100381610038161003816 (17)

Then

minusradic10

1198861198882atanh 119888 (119907 + 2119888)

|119888| radic1199072+ 119888119907 + 119888

2

= ℎ minus10038161003816100381610038161205851003816100381610038161003816 (18)

International Journal of Analysis 3

Solving (18) we obtain

119907 = (minus 2119888

1 minus tanh2 [

[

radic1198861198882

10

(10038161003816100381610038161205851003816100381610038161003816minus ℎ)

]

]

plusmn 2 |119888|(3

1 minus tanh2 [

[

radic1198861198882

10

(10038161003816100381610038161205851003816100381610038161003816minus ℎ)

]

]

times tanh2 [

[

radic1198861198882

10

(10038161003816100381610038161205851003816100381610038161003816minus ℎ)

]

]

)

12

)

times(1 minus 4tanh2 [

[

radic1198861198882

10

(10038161003816100381610038161205851003816100381610038161003816minus ℎ)

]

]

)

minus1

(19)

Therefore

119906 (119909 119905) = ( minus 2119888 [1 minus tanh2119891 (119909 minus 119888119905)]

+2 |119888|radic3 [1minustanh2119891 (119909 minus 119888119905)] tanh2119891 (119909 minus 119888119905))

times (1 minus 4tanh2119891 (119909 minus 119888119905))minus1

(20)

or

119906 (119909 119905) = ( minus 2119888 [1 minus tanh2119891 (119909 minus 119888119905)]

minus2 |119888|radic3 [1minustanh2119891 (119909 minus 119888119905)] tanh2119891 (119909 minus 119888119905))

times (1 minus 4tanh2119891 (119909 minus 119888119905))minus1

(21)

where 119891(119909 minus 119888119905) = radic(119886119888210)(|119909 minus 119888119905| minus ℎ)

3 Ansaumltz for the Generalized Camassa-HolmEquation

From [18] we have the following ordinary differential equa-tion in 119865

21198653 d2119865d1205852

+ 41198652(

d119865d120585)

2

=

119886

119898 + 1

119898+1

sum

119896=0

(

119898 + 1

119896)1198652119896119888119898+1minus119896

+ (2119896 minus 119888) 1198652+ 2119896119888 minus 119888

2+ 119887

(22)

where 119865 = radic119907 minus 119888

Let d119865d120585 = 119866 Then d2119865d1205852 = 119866(d119866d119865) Substitutingthis into (22) gives the following first order Bernoullirsquosordinary differential equation

d119866d119865+

2

119865

119866 = ((

119886

119898 + 1

119898+1

sum

119896=0

(

119898 + 1

119896)1198652119896119888119898+1minus119896

+ (2119896 minus 119888) 1198652+ 2119896119888 minus 119888

2+ 119887)

times (21198653)

minus1

)119866minus1

(23)

Solving (23) leads to

1198662= 119865minus4119892 + int[

119886

119898 + 1

119898+1

sum

119896=0

(

119898 + 1

119896)1198652119896119888119898+1minus119896

+ (2119896 minus 119888) 1198652+ 2119896119888 minus 119888

2+ 119887]119865d119865

(24)

Therefore

(119865

d119865d120585)

2

= 119892119865minus2+ [

119886

119898 + 1

119898+1

sum

119896=0

1

2119896 + 2

(

119898 + 1

119896)1198652119896119888119898+1minus119896

+

1

4

(2119896 minus 119888) 1198652+

1

2

(2119896119888 minus 1198882+ 119887) ]

(25)

where 119892 is an arbitrary constant We observe that 119865(d119865d120585)hardly becomes a polynomial in 119865 unless in the particularcases So this is a new ansatz comparedwith the ansatz in [18]Similarly we have

(119865

d119865d 10038161003816100381610038161205851003816100381610038161003816

)

2

= 119892119865minus2+ [

119886

119898 + 1

119898+1

sum

119896=0

1

2119896 + 2

(

119898 + 1

119896)1198652119896119888119898+1minus119896

+

1

4

(2119896 minus 119888) 1198652+

1

2

(2119896119888 minus 1198882+ 119887) ]

(26)

Seemingly (26) is more difficult than (8) On the contrary(26) is much easier to be solved than (8) This can be seen inthe following

When119898 = 1 let 119892 = 0 For 1198652 = 120579 we have

int(

119886

3

1205792+

1

4

(2119896 minus 119888 + 4119886119888) 120579

+

1

2

(2119896119888 minus 1198882+ 21198861198882+ 119887))

minus12

d120579 = 119903 minus 10038161003816100381610038161205851003816100381610038161003816

(27)

4 International Journal of Analysis

where 119903 is an arbitrary constant So

radic3

119886

ln[

[

(120579 +

3 (2119896 minus 119888 + 4119886119888)

8119886

)

+radic1205792+

3 (2119896 minus 119888 + 4119886119888)

4119886

120579+

3 (2119896119888 minus 1198882+ 21198861198882+ 119887)

2119886

]

]

= 119903 minus10038161003816100381610038161205851003816100381610038161003816

(28)

Solving (28) we have that when 3(2119896119888 minus 1198882 + 21198861198882 + 119887)(2119886) minus9(2119896 minus 119888 + 4119886119888)

2(161198862) gt 0

120579 =radic3 (2119896119888 minus 119888

2+ 21198861198882+ 119887)

2119886

minus

9(2119896 minus 119888 + 4119886119888)2

161198862

times sinh[radic3119886

(119903 minus10038161003816100381610038161205851003816100381610038161003816)] minus

3 (2119896 minus 119888 + 4119886119888)

8119886

(29)

when 3(2119896119888minus1198882+21198861198882+119887)(2119886)minus9(2119896minus119888+4119886119888)2(161198862) lt 0

120579 =radic3 (2119896119888 minus 119888

2+ 21198861198882+ 119887)

2119886

minus

9(2119896 minus 119888 + 4119886119888)2

161198862

times cosh[radic3119886

(119903 minus10038161003816100381610038161205851003816100381610038161003816)] minus

3 (2119896 minus 119888 + 4119886119888)

8119886

(30)

Equations (29) and (30) have no asymptotic behavior When3(2119896119888minus119888

2+21198861198882+119887)(2119886) = 9(2119896minus119888+4119886119888)

2(161198862) we obtain

120579 =

1

2

eradic(1198863)(119903minus|120585|) minus 3 (2119896 minus 119888 + 4119886119888)16119886

(31)

So

119906 (119909 119905) =

1

2

eradic(1198863)(119903minus|119909minus119888119905|) minus 6119896 minus 3119888 minus 411988611988816119886

(32)

These solitary solutions have the first derivative discontinuityat the wave peak

When119898 = 2 let 119892 = 0 Consider the following

int(

119886

6

1205793+

2119886119888

3

1205792+

1

4

(2119896 minus 119888 + 41198861198882) 120579

+

1

2

(2119896 minus 1198882+

41198861198883

3

+ 119887))

minus12

d120579 = 119903 minus 10038161003816100381610038161205851003816100381610038161003816

(33)

Take 2119896 minus 119888 + 41198861198882 = 0 and 2119896 minus 1198882 + 411988611988833 + 119887 = 0 then

minusradic6

119886119888

atanhradic120579 + 41198884119888

= 119903 minus10038161003816100381610038161205851003816100381610038161003816 (34)

From (34) we have

120579 = 4119888 tanh2 [radic1198861198886

(10038161003816100381610038161205851003816100381610038161003816minus 119903)] minus 4119888 (35)

Therefore we have the solitary wave solutions of the general-ized Camassa-Holm equation as

119906 (119909 119905) = 4119888 tanh2 [radic1198861198886

(|119909 minus 119888119905| minus 119903)] minus 3119888 (36)

Take 2119896minus119888+41198861198882 = 811988611988823 and 2119896minus1198882 +411988611988833+119887 = 0 then

radic12

119886119888

atanradic 1205792119888

= 119903 minus10038161003816100381610038161205851003816100381610038161003816 (37)

Similarly we have

119906 (119909 119905) = 2119888 tan2 [radic 11988611988812

(119903 minus |119909 minus 119888119905|)] + 119888 (38)

This result is the same as (28) in [18] But it is easily to obtainWhen119898 = 3 let 119892 = 0 Consider the following

int(

119886

10

1205794+

119886119888

2

1205793+ 11988611988821205792+

1

4

(2119896 minus 119888 + 41198861198883) 120579

+

1

2

(2119896 minus 1198882+ 1198861198884+ 119887))

minus12

d120579 = 119903 minus 10038161003816100381610038161205851003816100381610038161003816

(39)

From (39) and 119886 119888 gt 0 we have seen that 119865(d119865d120585) is not apolynomial in 119865 (compared with Equation (35) in [18]) Take2119896 minus 119888 + 4119886119888

3= 0 and 2119896 minus 1198882 + 1198861198884 + 119887 = 0 then

minusradic1

1198861198882atanh

radic10 (119888120579 + 41198882)

4 |119888| radic1205792+ 5119888120579 + 10119888

2

= 119903 minus10038161003816100381610038161205851003816100381610038161003816 (40)

Solving (40) we obtain

120579 = ( minus 20119888 [1 minus tanh2119901 (120585)]

plusmn 4 |119888| radic15 [1 minus tanh2119901 (120585)] tanh2119901 (120585))

times (5 minus 8tanh2119901 (120585))minus1

(41)

where 119901(120585) = radic119886|119888|(|120585| minus 119903) Therefore

119906 (119909 119905) = (( minus 20119888 [1 minus tanh2119901 (119909 minus 119888119905)] + 4 |119888|

times radic15 [1 minus tanh2119901 (119909 minus 119888119905)] tanh2119901 (119909 minus 119888119905))

times (5 minus 8tanh2119901 (119909 minus 119888119905))minus1

) + 119888

(42)or

119906 (119909 119905) = (( minus 20119888 [1 minus tanh2119901 (119909 minus 119888119905)] minus 4 |119888|

times radic15 [1 minus tanh2119901 (119909 minus 119888119905)] tanh2119901 (119909 minus 119888119905))

times (5 minus 8 tanh2119901 (119909 minus 119888119905))minus1

) + 119888

(43)

where 119901(119909 minus 119888119905) = radic119886|119888|(|119909 minus 119888119905| minus 119903)

International Journal of Analysis 5

4 Numerical Simulation Examples

Example 1 In (12) take 119888 = 1 ℎ = 0 and 119886 = 3 Then thesolitary wave solution is 119906(119909 119905) = 119890minus|119909minus119905|

We want to show figures with peakon feature But to savespace omitting figures we only give MATLAB program ldquo119909 =minus10 0001 10 119906 = exp(minusabs(119909)) plot(119909 119906)rdquo for 119905 = 0 andldquofor 119899 = 1 101 119909(119899) = (119899 minus 1)5 minus 10 119905(119899) = 119909(119899) end[119909(119899) 119905(119899)] = meshgrid(119909(119899) 119905(119899)) for 119899 = 1 101 for 119898 =

1 101 119906(119899119898) = exp(minusabs(119909(119899) minus 119905(119898))) end end mesh(119906)rdquofor the 3-dimensional case

Example 2 In (16) take 119888 = 1 ℎ = 0 and 119886 = 24 Thensolitary wave solution is 119906(119909 119905) = tanh2(|119909 minus 119905| minus 1) minus 1

The figures with peakon feature will be constructed byldquo119909 = minus10 0001 10 119906 = (tanh(abs(119909)minus1)) and2minus1 plot(119909 119906)rdquofor 119905 = 0 and ldquofor 119899 = 1 101 119909(119899) = (119899minus1)5minus10 119905(119899) = 119909(119899)end [119909(119899) 119905(119899)] = meshgrid(119909(119899) 119905(119899)) for 119899 = 1 101 for119898 = 1 101 119906(119899119898) = (tanh(abs(119909(119899) minus 119905(119898)) minus 1)) and2 minus 1end end mesh(119906)rdquo for the 3-dimensional case

Example 3 In (20) take 119888 = 1 ℎ = 1 and 119886 = 10 Thensymmetrically solitary wave solution is

119906 (119909 119905)

= minus 2 [1 minus tanh2 (|119909 minus 119905| minus 1)]

+2radic3 [1 minus tanh2 (|119909 minus 119905| minus 1)] tanh2 (|119909 minus 119905| minus 1)

times (1 minus 4tanh2 (|119909 minus 119905| minus 1))minus1

(44)

For 119905 = 0 when (119886 = minus2 119887 = 2) (119886 = 231 119887 = 232) and(119886 = 232 119887 = 5) respectively the figures are with peakonfeature 119909 = a 00001 b119910 = (minus2lowast(1minustanh(abs(119909)minus1) and2)+2lowastsqrt(3lowast(1minustanh(abs(119909)minus1) and2)lowasttanh(abs(119909))))(1minus4lowast(1minustanh(abs(119909)minus1) and2)) plot(119909 119910) Replace 119909 by 119909(119899)minus119905(119898)and the others are similar to Example 2 program for the 3-dimensional case

5 Conclusions

In this paper we make simplification of the nonlineargeneralized Camassa-Holm equation and give the improvedansatz for the generalizedCamassa-Holm equationThe threenumerical simulation examples with mATLAB programsdemonstrate solitary wave solutions with peakon featurewhich show our method applicable This method may beapplied to many other nonlinear equations

Acknowledgments

This work was supported partially by National NaturalScience Foundation of China (Grant nos 10871056 and10971150) and by Science Research Foundation in HarbinInstitute of Technology (Grant no HITC200708)

References

[1] R Camassa and D D Holm ldquoAn integrable shallow waterequation with peaked solitonsrdquo Physical Review Letters vol 71no 11 pp 1661ndash1664 1993

[2] Y Long Z Li and W Rui ldquoNew travelling wave solutionsfor a nonlinearly dispersive wave equation of Camassa-Holmequation typerdquo Applied Mathematics and Computation vol 217no 4 pp 1315ndash1320 2010

[3] Y Xu and L Tian ldquoAttractor for a coupled nonhomogeneousCamassa-Holm equationrdquo International Journal of NonlinearScience vol 9 no 1 pp 118ndash122 2010

[4] M Song and Z R Liu ldquoTraveling wave solutions for thegeneralized Zakharov equationsrdquo Mathematical Problems inEngineering vol 2012 Article ID 747295 14 pages 2012

[5] S Tang Y Xiao and Z Wang ldquoTravelling wave solutionsfor a class of nonlinear fourth order variant of a generalizedCamassa-Holm equationrdquo Applied Mathematics and Computa-tion vol 210 no 1 pp 39ndash47 2009

[6] B He W Rui C Chen and S Li ldquoExact travelling wavesolutions of a generalized Camassa-Holm equation using theintegral bifurcationmethodrdquoAppliedMathematics and Compu-tation vol 206 no 1 pp 141ndash149 2008

[7] A-M Wazwaz ldquoNew solitary wave solutions to the modifiedforms of Degasperis-Procesi and Camassa-Holm equationsrdquoApplied Mathematics and Computation vol 186 no 1 pp 130ndash141 2007

[8] A-M Wazwaz ldquoPeakons kinks compactons and solitarypatterns solutions for a family of Camassa-Holm equationsby using new hyperbolic schemesrdquo Applied Mathematics andComputation vol 182 no 1 pp 412ndash424 2006

[9] S Chen C Foias D DHolm E Olson E S Titi and SWynneldquoCamassa-Holm equations as a closure model for turbulentchannel and pipe flowrdquo Physical Review Letters vol 81 no 24pp 5338ndash5341 1998

[10] F Cooper and H Shepard ldquoSolitons in the Camassa-Holmshallow water equationrdquo Physics Letters A vol 194 no 4 pp246ndash250 1994

[11] M Fisher and J Schiff ldquoThe Camassa Holm equation con-served quantities and the initial value problemrdquo Physics LettersA vol 259 no 5 pp 371ndash376 1999

[12] Z Liu T Qian and M Tang ldquoPeakons of the Camassa-Holmequationrdquo Applied Mathematical Modelling vol 26 pp 473ndash480 2002

[13] Z Liu and T Qian ldquoPeakons and their bifurcation in ageneralized Camassa-Holm equationrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol11 no 3 pp 781ndash792 2001

[14] T Qian and M Tang ldquoPeakons and periodic cusp waves ina generalized Camassa-Holm equationrdquo Chaos Solitons andFractals vol 12 no 7 pp 1347ndash1360 2001

[15] R A Kraenkel M Senthilvelan and A I Zenchuk ldquoOnthe integrable perturbations of the Camassa-Holm equationrdquoJournal of Mathematical Physics vol 41 no 5 pp 3160ndash31692000

[16] L Tian and J Yin ldquoNew compacton solutions and solitarywave solutions of fully nonlinear generalized Camassa-Holmequationsrdquo Chaos Solitons and Fractals vol 20 no 2 pp 289ndash299 2004

6 International Journal of Analysis

[17] Z Liu T Jiang P Qin and Q Xu ldquoTrigonometric functionperiodic wave solutions and their limit forms for the KdV-like and the PC-like equationsrdquo Mathematical Problems inEngineering vol 2011 Article ID 810217 23 pages 2011

[18] S A Khuri ldquoNew ansatz for obtaining wave solutions ofthe generalized Camassa-Holm equationrdquo Chaos Solitons ampFractals vol 25 no 3 pp 705ndash710 2005

[19] L Tian and X Song ldquoNew peaked solitary wave solutions ofthe generalized Camassa-Holm equationrdquo Chaos Solitons andFractals vol 19 no 3 pp 621ndash637 2004

[20] R Camassa andA I Zenchuk ldquoOn the initial value problem fora completely integrable shallow water wave equationrdquo PhysicsLetters A vol 281 no 1 pp 26ndash33 2001

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

2 International Journal of Analysis

Integrating both sides of (3) we obtain

(119907 minus 119888)

d2119907d1205852

+

1

2

(

d119907d120585)

2

=

119886

119898 + 1

119907119898+1+ (2119896 minus 119888) 119907 + 119887 (4)

where 119887 is an arbitrary constant Let d119907d120585 = 119908 Thend2119907d1205852 = 119908(d119908d119907) Substituting this into (4) yields

d119908d119907+

1

2 (119907 minus 119888)

119908 =

(119886 (119898 + 1)) 119907119898+1+ (2119896 minus 119888) 119907 + 119887

119907 minus 119888

119908minus1

(5)

Solving (4) leads to

1199082= (119907 minus 119888)

minus1119889 + int 2 [

119886

119898 + 1

119907119898+1+ (2119896 minus 119888) 119907 + 119887] d119907

(6)

where 119889 is an arbitrary constant Therefore

(

d119907d120585)

2

= (119907 minus 119888)minus1

times 119889 + [

2119886

(119898 + 1) (119898 + 2)

119907119898+2+ (2119896 minus 119888) 119907

2+ 119887119907]

(7)

From (d120585)2 = (d|120585|)2 we have

(

d119907d 10038161003816100381610038161205851003816100381610038161003816

)

2

= (119907 minus 119888)minus1

times 119889 + [

2119886

(119898 + 1) (119898 + 2)

119907119898+2+ (2119896 minus 119888) 119907

2+ 119887119907]

(8)

Therefore

intradic

119907 minus 119888

119889 + [(2119886 ((119898 + 1) (119898 + 2))) 119907119898+2+ (2119896 minus 119888) 119907

2+ 119887119907]

d119907

=

ℎ + 120585

ℎ minus 120585

ℎ +10038161003816100381610038161205851003816100381610038161003816

ℎ minus10038161003816100381610038161205851003816100381610038161003816

(9)

where ℎ is an arbitrary constantWhen 119898 = 1 (9) is the case of the Camassa-Holm

equation We take 2119896 = 119888 119886119888 minus 3119888 + 2119896 = 0 119889 = 0 and 119887 = 0then

radic3

119886

ln 119888119907

=

ℎ + 120585

ℎ minus 120585

ℎ +10038161003816100381610038161205851003816100381610038161003816

ℎ minus10038161003816100381610038161205851003816100381610038161003816

(10)

So

119907 =

119888eradic(1198863)(ℎ+120585)119888eradic(1198863)(ℎminus120585)119888eradic(1198863)(ℎ+|120585|)119888eradic(1198863)(ℎminus|120585|)

(11)

It can be checked that 119906(119909 119905) = 119888eℎ+radic(1198863)(119909minus119888119905) 119906(119909 119905) =119888eℎminusradic(1198863)(119909minus119888119905) and 119906(119909 119905) = 119888eℎ+radic(1198863)|119909minus119888119905| are infinitely greatsolutions of the Camassa-Holm equation without asymptoticbehavior [20] Only

119906 (119909 119905) = 119888eℎminusradic(1198863)|119909minus119888119905| (12)

are the solitary wave solutions of the Camassa-Holm equa-tion

When119898 = 2 we take 2119896 lt 119888 1198861198882minus6119888+12119896 = 0 119889 = 0 and119887 = 0 In the similar way in the case119898 = 1 we only choose

radic6

119886

int

1

119907radic119907 + 119888

d120593 = ℎ minus 10038161003816100381610038161205851003816100381610038161003816 (13)

then

minusradic24

119886119888

atanhradic119907 + 119888119888

= ℎ minus10038161003816100381610038161205851003816100381610038161003816 (14)

From (14) we have

119907 = 119888 tanh2 [radic 11988611988824

(10038161003816100381610038161205851003816100381610038161003816minus ℎ)] minus 119888 (15)

Therefore we have the solitary wave solutions of the general-ized Camassa-Holm equation as

119906 (119909 119905) = 119888 tanh2 [radic 11988611988824

(|119909 minus 119888119905| minus ℎ)] minus 119888 (16)

When 119898 = 3 we take 2119896 = 119888 1198861198883 minus 10119888 + 20119896 = 0 119889 = 0and 119887 = 0 then

radic10

119886

int

1

radic1199072+ 119888119907 + 119888

2

d119907 = ℎ minus 10038161003816100381610038161205851003816100381610038161003816 (17)

Then

minusradic10

1198861198882atanh 119888 (119907 + 2119888)

|119888| radic1199072+ 119888119907 + 119888

2

= ℎ minus10038161003816100381610038161205851003816100381610038161003816 (18)

International Journal of Analysis 3

Solving (18) we obtain

119907 = (minus 2119888

1 minus tanh2 [

[

radic1198861198882

10

(10038161003816100381610038161205851003816100381610038161003816minus ℎ)

]

]

plusmn 2 |119888|(3

1 minus tanh2 [

[

radic1198861198882

10

(10038161003816100381610038161205851003816100381610038161003816minus ℎ)

]

]

times tanh2 [

[

radic1198861198882

10

(10038161003816100381610038161205851003816100381610038161003816minus ℎ)

]

]

)

12

)

times(1 minus 4tanh2 [

[

radic1198861198882

10

(10038161003816100381610038161205851003816100381610038161003816minus ℎ)

]

]

)

minus1

(19)

Therefore

119906 (119909 119905) = ( minus 2119888 [1 minus tanh2119891 (119909 minus 119888119905)]

+2 |119888|radic3 [1minustanh2119891 (119909 minus 119888119905)] tanh2119891 (119909 minus 119888119905))

times (1 minus 4tanh2119891 (119909 minus 119888119905))minus1

(20)

or

119906 (119909 119905) = ( minus 2119888 [1 minus tanh2119891 (119909 minus 119888119905)]

minus2 |119888|radic3 [1minustanh2119891 (119909 minus 119888119905)] tanh2119891 (119909 minus 119888119905))

times (1 minus 4tanh2119891 (119909 minus 119888119905))minus1

(21)

where 119891(119909 minus 119888119905) = radic(119886119888210)(|119909 minus 119888119905| minus ℎ)

3 Ansaumltz for the Generalized Camassa-HolmEquation

From [18] we have the following ordinary differential equa-tion in 119865

21198653 d2119865d1205852

+ 41198652(

d119865d120585)

2

=

119886

119898 + 1

119898+1

sum

119896=0

(

119898 + 1

119896)1198652119896119888119898+1minus119896

+ (2119896 minus 119888) 1198652+ 2119896119888 minus 119888

2+ 119887

(22)

where 119865 = radic119907 minus 119888

Let d119865d120585 = 119866 Then d2119865d1205852 = 119866(d119866d119865) Substitutingthis into (22) gives the following first order Bernoullirsquosordinary differential equation

d119866d119865+

2

119865

119866 = ((

119886

119898 + 1

119898+1

sum

119896=0

(

119898 + 1

119896)1198652119896119888119898+1minus119896

+ (2119896 minus 119888) 1198652+ 2119896119888 minus 119888

2+ 119887)

times (21198653)

minus1

)119866minus1

(23)

Solving (23) leads to

1198662= 119865minus4119892 + int[

119886

119898 + 1

119898+1

sum

119896=0

(

119898 + 1

119896)1198652119896119888119898+1minus119896

+ (2119896 minus 119888) 1198652+ 2119896119888 minus 119888

2+ 119887]119865d119865

(24)

Therefore

(119865

d119865d120585)

2

= 119892119865minus2+ [

119886

119898 + 1

119898+1

sum

119896=0

1

2119896 + 2

(

119898 + 1

119896)1198652119896119888119898+1minus119896

+

1

4

(2119896 minus 119888) 1198652+

1

2

(2119896119888 minus 1198882+ 119887) ]

(25)

where 119892 is an arbitrary constant We observe that 119865(d119865d120585)hardly becomes a polynomial in 119865 unless in the particularcases So this is a new ansatz comparedwith the ansatz in [18]Similarly we have

(119865

d119865d 10038161003816100381610038161205851003816100381610038161003816

)

2

= 119892119865minus2+ [

119886

119898 + 1

119898+1

sum

119896=0

1

2119896 + 2

(

119898 + 1

119896)1198652119896119888119898+1minus119896

+

1

4

(2119896 minus 119888) 1198652+

1

2

(2119896119888 minus 1198882+ 119887) ]

(26)

Seemingly (26) is more difficult than (8) On the contrary(26) is much easier to be solved than (8) This can be seen inthe following

When119898 = 1 let 119892 = 0 For 1198652 = 120579 we have

int(

119886

3

1205792+

1

4

(2119896 minus 119888 + 4119886119888) 120579

+

1

2

(2119896119888 minus 1198882+ 21198861198882+ 119887))

minus12

d120579 = 119903 minus 10038161003816100381610038161205851003816100381610038161003816

(27)

4 International Journal of Analysis

where 119903 is an arbitrary constant So

radic3

119886

ln[

[

(120579 +

3 (2119896 minus 119888 + 4119886119888)

8119886

)

+radic1205792+

3 (2119896 minus 119888 + 4119886119888)

4119886

120579+

3 (2119896119888 minus 1198882+ 21198861198882+ 119887)

2119886

]

]

= 119903 minus10038161003816100381610038161205851003816100381610038161003816

(28)

Solving (28) we have that when 3(2119896119888 minus 1198882 + 21198861198882 + 119887)(2119886) minus9(2119896 minus 119888 + 4119886119888)

2(161198862) gt 0

120579 =radic3 (2119896119888 minus 119888

2+ 21198861198882+ 119887)

2119886

minus

9(2119896 minus 119888 + 4119886119888)2

161198862

times sinh[radic3119886

(119903 minus10038161003816100381610038161205851003816100381610038161003816)] minus

3 (2119896 minus 119888 + 4119886119888)

8119886

(29)

when 3(2119896119888minus1198882+21198861198882+119887)(2119886)minus9(2119896minus119888+4119886119888)2(161198862) lt 0

120579 =radic3 (2119896119888 minus 119888

2+ 21198861198882+ 119887)

2119886

minus

9(2119896 minus 119888 + 4119886119888)2

161198862

times cosh[radic3119886

(119903 minus10038161003816100381610038161205851003816100381610038161003816)] minus

3 (2119896 minus 119888 + 4119886119888)

8119886

(30)

Equations (29) and (30) have no asymptotic behavior When3(2119896119888minus119888

2+21198861198882+119887)(2119886) = 9(2119896minus119888+4119886119888)

2(161198862) we obtain

120579 =

1

2

eradic(1198863)(119903minus|120585|) minus 3 (2119896 minus 119888 + 4119886119888)16119886

(31)

So

119906 (119909 119905) =

1

2

eradic(1198863)(119903minus|119909minus119888119905|) minus 6119896 minus 3119888 minus 411988611988816119886

(32)

These solitary solutions have the first derivative discontinuityat the wave peak

When119898 = 2 let 119892 = 0 Consider the following

int(

119886

6

1205793+

2119886119888

3

1205792+

1

4

(2119896 minus 119888 + 41198861198882) 120579

+

1

2

(2119896 minus 1198882+

41198861198883

3

+ 119887))

minus12

d120579 = 119903 minus 10038161003816100381610038161205851003816100381610038161003816

(33)

Take 2119896 minus 119888 + 41198861198882 = 0 and 2119896 minus 1198882 + 411988611988833 + 119887 = 0 then

minusradic6

119886119888

atanhradic120579 + 41198884119888

= 119903 minus10038161003816100381610038161205851003816100381610038161003816 (34)

From (34) we have

120579 = 4119888 tanh2 [radic1198861198886

(10038161003816100381610038161205851003816100381610038161003816minus 119903)] minus 4119888 (35)

Therefore we have the solitary wave solutions of the general-ized Camassa-Holm equation as

119906 (119909 119905) = 4119888 tanh2 [radic1198861198886

(|119909 minus 119888119905| minus 119903)] minus 3119888 (36)

Take 2119896minus119888+41198861198882 = 811988611988823 and 2119896minus1198882 +411988611988833+119887 = 0 then

radic12

119886119888

atanradic 1205792119888

= 119903 minus10038161003816100381610038161205851003816100381610038161003816 (37)

Similarly we have

119906 (119909 119905) = 2119888 tan2 [radic 11988611988812

(119903 minus |119909 minus 119888119905|)] + 119888 (38)

This result is the same as (28) in [18] But it is easily to obtainWhen119898 = 3 let 119892 = 0 Consider the following

int(

119886

10

1205794+

119886119888

2

1205793+ 11988611988821205792+

1

4

(2119896 minus 119888 + 41198861198883) 120579

+

1

2

(2119896 minus 1198882+ 1198861198884+ 119887))

minus12

d120579 = 119903 minus 10038161003816100381610038161205851003816100381610038161003816

(39)

From (39) and 119886 119888 gt 0 we have seen that 119865(d119865d120585) is not apolynomial in 119865 (compared with Equation (35) in [18]) Take2119896 minus 119888 + 4119886119888

3= 0 and 2119896 minus 1198882 + 1198861198884 + 119887 = 0 then

minusradic1

1198861198882atanh

radic10 (119888120579 + 41198882)

4 |119888| radic1205792+ 5119888120579 + 10119888

2

= 119903 minus10038161003816100381610038161205851003816100381610038161003816 (40)

Solving (40) we obtain

120579 = ( minus 20119888 [1 minus tanh2119901 (120585)]

plusmn 4 |119888| radic15 [1 minus tanh2119901 (120585)] tanh2119901 (120585))

times (5 minus 8tanh2119901 (120585))minus1

(41)

where 119901(120585) = radic119886|119888|(|120585| minus 119903) Therefore

119906 (119909 119905) = (( minus 20119888 [1 minus tanh2119901 (119909 minus 119888119905)] + 4 |119888|

times radic15 [1 minus tanh2119901 (119909 minus 119888119905)] tanh2119901 (119909 minus 119888119905))

times (5 minus 8tanh2119901 (119909 minus 119888119905))minus1

) + 119888

(42)or

119906 (119909 119905) = (( minus 20119888 [1 minus tanh2119901 (119909 minus 119888119905)] minus 4 |119888|

times radic15 [1 minus tanh2119901 (119909 minus 119888119905)] tanh2119901 (119909 minus 119888119905))

times (5 minus 8 tanh2119901 (119909 minus 119888119905))minus1

) + 119888

(43)

where 119901(119909 minus 119888119905) = radic119886|119888|(|119909 minus 119888119905| minus 119903)

International Journal of Analysis 5

4 Numerical Simulation Examples

Example 1 In (12) take 119888 = 1 ℎ = 0 and 119886 = 3 Then thesolitary wave solution is 119906(119909 119905) = 119890minus|119909minus119905|

We want to show figures with peakon feature But to savespace omitting figures we only give MATLAB program ldquo119909 =minus10 0001 10 119906 = exp(minusabs(119909)) plot(119909 119906)rdquo for 119905 = 0 andldquofor 119899 = 1 101 119909(119899) = (119899 minus 1)5 minus 10 119905(119899) = 119909(119899) end[119909(119899) 119905(119899)] = meshgrid(119909(119899) 119905(119899)) for 119899 = 1 101 for 119898 =

1 101 119906(119899119898) = exp(minusabs(119909(119899) minus 119905(119898))) end end mesh(119906)rdquofor the 3-dimensional case

Example 2 In (16) take 119888 = 1 ℎ = 0 and 119886 = 24 Thensolitary wave solution is 119906(119909 119905) = tanh2(|119909 minus 119905| minus 1) minus 1

The figures with peakon feature will be constructed byldquo119909 = minus10 0001 10 119906 = (tanh(abs(119909)minus1)) and2minus1 plot(119909 119906)rdquofor 119905 = 0 and ldquofor 119899 = 1 101 119909(119899) = (119899minus1)5minus10 119905(119899) = 119909(119899)end [119909(119899) 119905(119899)] = meshgrid(119909(119899) 119905(119899)) for 119899 = 1 101 for119898 = 1 101 119906(119899119898) = (tanh(abs(119909(119899) minus 119905(119898)) minus 1)) and2 minus 1end end mesh(119906)rdquo for the 3-dimensional case

Example 3 In (20) take 119888 = 1 ℎ = 1 and 119886 = 10 Thensymmetrically solitary wave solution is

119906 (119909 119905)

= minus 2 [1 minus tanh2 (|119909 minus 119905| minus 1)]

+2radic3 [1 minus tanh2 (|119909 minus 119905| minus 1)] tanh2 (|119909 minus 119905| minus 1)

times (1 minus 4tanh2 (|119909 minus 119905| minus 1))minus1

(44)

For 119905 = 0 when (119886 = minus2 119887 = 2) (119886 = 231 119887 = 232) and(119886 = 232 119887 = 5) respectively the figures are with peakonfeature 119909 = a 00001 b119910 = (minus2lowast(1minustanh(abs(119909)minus1) and2)+2lowastsqrt(3lowast(1minustanh(abs(119909)minus1) and2)lowasttanh(abs(119909))))(1minus4lowast(1minustanh(abs(119909)minus1) and2)) plot(119909 119910) Replace 119909 by 119909(119899)minus119905(119898)and the others are similar to Example 2 program for the 3-dimensional case

5 Conclusions

In this paper we make simplification of the nonlineargeneralized Camassa-Holm equation and give the improvedansatz for the generalizedCamassa-Holm equationThe threenumerical simulation examples with mATLAB programsdemonstrate solitary wave solutions with peakon featurewhich show our method applicable This method may beapplied to many other nonlinear equations

Acknowledgments

This work was supported partially by National NaturalScience Foundation of China (Grant nos 10871056 and10971150) and by Science Research Foundation in HarbinInstitute of Technology (Grant no HITC200708)

References

[1] R Camassa and D D Holm ldquoAn integrable shallow waterequation with peaked solitonsrdquo Physical Review Letters vol 71no 11 pp 1661ndash1664 1993

[2] Y Long Z Li and W Rui ldquoNew travelling wave solutionsfor a nonlinearly dispersive wave equation of Camassa-Holmequation typerdquo Applied Mathematics and Computation vol 217no 4 pp 1315ndash1320 2010

[3] Y Xu and L Tian ldquoAttractor for a coupled nonhomogeneousCamassa-Holm equationrdquo International Journal of NonlinearScience vol 9 no 1 pp 118ndash122 2010

[4] M Song and Z R Liu ldquoTraveling wave solutions for thegeneralized Zakharov equationsrdquo Mathematical Problems inEngineering vol 2012 Article ID 747295 14 pages 2012

[5] S Tang Y Xiao and Z Wang ldquoTravelling wave solutionsfor a class of nonlinear fourth order variant of a generalizedCamassa-Holm equationrdquo Applied Mathematics and Computa-tion vol 210 no 1 pp 39ndash47 2009

[6] B He W Rui C Chen and S Li ldquoExact travelling wavesolutions of a generalized Camassa-Holm equation using theintegral bifurcationmethodrdquoAppliedMathematics and Compu-tation vol 206 no 1 pp 141ndash149 2008

[7] A-M Wazwaz ldquoNew solitary wave solutions to the modifiedforms of Degasperis-Procesi and Camassa-Holm equationsrdquoApplied Mathematics and Computation vol 186 no 1 pp 130ndash141 2007

[8] A-M Wazwaz ldquoPeakons kinks compactons and solitarypatterns solutions for a family of Camassa-Holm equationsby using new hyperbolic schemesrdquo Applied Mathematics andComputation vol 182 no 1 pp 412ndash424 2006

[9] S Chen C Foias D DHolm E Olson E S Titi and SWynneldquoCamassa-Holm equations as a closure model for turbulentchannel and pipe flowrdquo Physical Review Letters vol 81 no 24pp 5338ndash5341 1998

[10] F Cooper and H Shepard ldquoSolitons in the Camassa-Holmshallow water equationrdquo Physics Letters A vol 194 no 4 pp246ndash250 1994

[11] M Fisher and J Schiff ldquoThe Camassa Holm equation con-served quantities and the initial value problemrdquo Physics LettersA vol 259 no 5 pp 371ndash376 1999

[12] Z Liu T Qian and M Tang ldquoPeakons of the Camassa-Holmequationrdquo Applied Mathematical Modelling vol 26 pp 473ndash480 2002

[13] Z Liu and T Qian ldquoPeakons and their bifurcation in ageneralized Camassa-Holm equationrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol11 no 3 pp 781ndash792 2001

[14] T Qian and M Tang ldquoPeakons and periodic cusp waves ina generalized Camassa-Holm equationrdquo Chaos Solitons andFractals vol 12 no 7 pp 1347ndash1360 2001

[15] R A Kraenkel M Senthilvelan and A I Zenchuk ldquoOnthe integrable perturbations of the Camassa-Holm equationrdquoJournal of Mathematical Physics vol 41 no 5 pp 3160ndash31692000

[16] L Tian and J Yin ldquoNew compacton solutions and solitarywave solutions of fully nonlinear generalized Camassa-Holmequationsrdquo Chaos Solitons and Fractals vol 20 no 2 pp 289ndash299 2004

6 International Journal of Analysis

[17] Z Liu T Jiang P Qin and Q Xu ldquoTrigonometric functionperiodic wave solutions and their limit forms for the KdV-like and the PC-like equationsrdquo Mathematical Problems inEngineering vol 2011 Article ID 810217 23 pages 2011

[18] S A Khuri ldquoNew ansatz for obtaining wave solutions ofthe generalized Camassa-Holm equationrdquo Chaos Solitons ampFractals vol 25 no 3 pp 705ndash710 2005

[19] L Tian and X Song ldquoNew peaked solitary wave solutions ofthe generalized Camassa-Holm equationrdquo Chaos Solitons andFractals vol 19 no 3 pp 621ndash637 2004

[20] R Camassa andA I Zenchuk ldquoOn the initial value problem fora completely integrable shallow water wave equationrdquo PhysicsLetters A vol 281 no 1 pp 26ndash33 2001

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

International Journal of Analysis 3

Solving (18) we obtain

119907 = (minus 2119888

1 minus tanh2 [

[

radic1198861198882

10

(10038161003816100381610038161205851003816100381610038161003816minus ℎ)

]

]

plusmn 2 |119888|(3

1 minus tanh2 [

[

radic1198861198882

10

(10038161003816100381610038161205851003816100381610038161003816minus ℎ)

]

]

times tanh2 [

[

radic1198861198882

10

(10038161003816100381610038161205851003816100381610038161003816minus ℎ)

]

]

)

12

)

times(1 minus 4tanh2 [

[

radic1198861198882

10

(10038161003816100381610038161205851003816100381610038161003816minus ℎ)

]

]

)

minus1

(19)

Therefore

119906 (119909 119905) = ( minus 2119888 [1 minus tanh2119891 (119909 minus 119888119905)]

+2 |119888|radic3 [1minustanh2119891 (119909 minus 119888119905)] tanh2119891 (119909 minus 119888119905))

times (1 minus 4tanh2119891 (119909 minus 119888119905))minus1

(20)

or

119906 (119909 119905) = ( minus 2119888 [1 minus tanh2119891 (119909 minus 119888119905)]

minus2 |119888|radic3 [1minustanh2119891 (119909 minus 119888119905)] tanh2119891 (119909 minus 119888119905))

times (1 minus 4tanh2119891 (119909 minus 119888119905))minus1

(21)

where 119891(119909 minus 119888119905) = radic(119886119888210)(|119909 minus 119888119905| minus ℎ)

3 Ansaumltz for the Generalized Camassa-HolmEquation

From [18] we have the following ordinary differential equa-tion in 119865

21198653 d2119865d1205852

+ 41198652(

d119865d120585)

2

=

119886

119898 + 1

119898+1

sum

119896=0

(

119898 + 1

119896)1198652119896119888119898+1minus119896

+ (2119896 minus 119888) 1198652+ 2119896119888 minus 119888

2+ 119887

(22)

where 119865 = radic119907 minus 119888

Let d119865d120585 = 119866 Then d2119865d1205852 = 119866(d119866d119865) Substitutingthis into (22) gives the following first order Bernoullirsquosordinary differential equation

d119866d119865+

2

119865

119866 = ((

119886

119898 + 1

119898+1

sum

119896=0

(

119898 + 1

119896)1198652119896119888119898+1minus119896

+ (2119896 minus 119888) 1198652+ 2119896119888 minus 119888

2+ 119887)

times (21198653)

minus1

)119866minus1

(23)

Solving (23) leads to

1198662= 119865minus4119892 + int[

119886

119898 + 1

119898+1

sum

119896=0

(

119898 + 1

119896)1198652119896119888119898+1minus119896

+ (2119896 minus 119888) 1198652+ 2119896119888 minus 119888

2+ 119887]119865d119865

(24)

Therefore

(119865

d119865d120585)

2

= 119892119865minus2+ [

119886

119898 + 1

119898+1

sum

119896=0

1

2119896 + 2

(

119898 + 1

119896)1198652119896119888119898+1minus119896

+

1

4

(2119896 minus 119888) 1198652+

1

2

(2119896119888 minus 1198882+ 119887) ]

(25)

where 119892 is an arbitrary constant We observe that 119865(d119865d120585)hardly becomes a polynomial in 119865 unless in the particularcases So this is a new ansatz comparedwith the ansatz in [18]Similarly we have

(119865

d119865d 10038161003816100381610038161205851003816100381610038161003816

)

2

= 119892119865minus2+ [

119886

119898 + 1

119898+1

sum

119896=0

1

2119896 + 2

(

119898 + 1

119896)1198652119896119888119898+1minus119896

+

1

4

(2119896 minus 119888) 1198652+

1

2

(2119896119888 minus 1198882+ 119887) ]

(26)

Seemingly (26) is more difficult than (8) On the contrary(26) is much easier to be solved than (8) This can be seen inthe following

When119898 = 1 let 119892 = 0 For 1198652 = 120579 we have

int(

119886

3

1205792+

1

4

(2119896 minus 119888 + 4119886119888) 120579

+

1

2

(2119896119888 minus 1198882+ 21198861198882+ 119887))

minus12

d120579 = 119903 minus 10038161003816100381610038161205851003816100381610038161003816

(27)

4 International Journal of Analysis

where 119903 is an arbitrary constant So

radic3

119886

ln[

[

(120579 +

3 (2119896 minus 119888 + 4119886119888)

8119886

)

+radic1205792+

3 (2119896 minus 119888 + 4119886119888)

4119886

120579+

3 (2119896119888 minus 1198882+ 21198861198882+ 119887)

2119886

]

]

= 119903 minus10038161003816100381610038161205851003816100381610038161003816

(28)

Solving (28) we have that when 3(2119896119888 minus 1198882 + 21198861198882 + 119887)(2119886) minus9(2119896 minus 119888 + 4119886119888)

2(161198862) gt 0

120579 =radic3 (2119896119888 minus 119888

2+ 21198861198882+ 119887)

2119886

minus

9(2119896 minus 119888 + 4119886119888)2

161198862

times sinh[radic3119886

(119903 minus10038161003816100381610038161205851003816100381610038161003816)] minus

3 (2119896 minus 119888 + 4119886119888)

8119886

(29)

when 3(2119896119888minus1198882+21198861198882+119887)(2119886)minus9(2119896minus119888+4119886119888)2(161198862) lt 0

120579 =radic3 (2119896119888 minus 119888

2+ 21198861198882+ 119887)

2119886

minus

9(2119896 minus 119888 + 4119886119888)2

161198862

times cosh[radic3119886

(119903 minus10038161003816100381610038161205851003816100381610038161003816)] minus

3 (2119896 minus 119888 + 4119886119888)

8119886

(30)

Equations (29) and (30) have no asymptotic behavior When3(2119896119888minus119888

2+21198861198882+119887)(2119886) = 9(2119896minus119888+4119886119888)

2(161198862) we obtain

120579 =

1

2

eradic(1198863)(119903minus|120585|) minus 3 (2119896 minus 119888 + 4119886119888)16119886

(31)

So

119906 (119909 119905) =

1

2

eradic(1198863)(119903minus|119909minus119888119905|) minus 6119896 minus 3119888 minus 411988611988816119886

(32)

These solitary solutions have the first derivative discontinuityat the wave peak

When119898 = 2 let 119892 = 0 Consider the following

int(

119886

6

1205793+

2119886119888

3

1205792+

1

4

(2119896 minus 119888 + 41198861198882) 120579

+

1

2

(2119896 minus 1198882+

41198861198883

3

+ 119887))

minus12

d120579 = 119903 minus 10038161003816100381610038161205851003816100381610038161003816

(33)

Take 2119896 minus 119888 + 41198861198882 = 0 and 2119896 minus 1198882 + 411988611988833 + 119887 = 0 then

minusradic6

119886119888

atanhradic120579 + 41198884119888

= 119903 minus10038161003816100381610038161205851003816100381610038161003816 (34)

From (34) we have

120579 = 4119888 tanh2 [radic1198861198886

(10038161003816100381610038161205851003816100381610038161003816minus 119903)] minus 4119888 (35)

Therefore we have the solitary wave solutions of the general-ized Camassa-Holm equation as

119906 (119909 119905) = 4119888 tanh2 [radic1198861198886

(|119909 minus 119888119905| minus 119903)] minus 3119888 (36)

Take 2119896minus119888+41198861198882 = 811988611988823 and 2119896minus1198882 +411988611988833+119887 = 0 then

radic12

119886119888

atanradic 1205792119888

= 119903 minus10038161003816100381610038161205851003816100381610038161003816 (37)

Similarly we have

119906 (119909 119905) = 2119888 tan2 [radic 11988611988812

(119903 minus |119909 minus 119888119905|)] + 119888 (38)

This result is the same as (28) in [18] But it is easily to obtainWhen119898 = 3 let 119892 = 0 Consider the following

int(

119886

10

1205794+

119886119888

2

1205793+ 11988611988821205792+

1

4

(2119896 minus 119888 + 41198861198883) 120579

+

1

2

(2119896 minus 1198882+ 1198861198884+ 119887))

minus12

d120579 = 119903 minus 10038161003816100381610038161205851003816100381610038161003816

(39)

From (39) and 119886 119888 gt 0 we have seen that 119865(d119865d120585) is not apolynomial in 119865 (compared with Equation (35) in [18]) Take2119896 minus 119888 + 4119886119888

3= 0 and 2119896 minus 1198882 + 1198861198884 + 119887 = 0 then

minusradic1

1198861198882atanh

radic10 (119888120579 + 41198882)

4 |119888| radic1205792+ 5119888120579 + 10119888

2

= 119903 minus10038161003816100381610038161205851003816100381610038161003816 (40)

Solving (40) we obtain

120579 = ( minus 20119888 [1 minus tanh2119901 (120585)]

plusmn 4 |119888| radic15 [1 minus tanh2119901 (120585)] tanh2119901 (120585))

times (5 minus 8tanh2119901 (120585))minus1

(41)

where 119901(120585) = radic119886|119888|(|120585| minus 119903) Therefore

119906 (119909 119905) = (( minus 20119888 [1 minus tanh2119901 (119909 minus 119888119905)] + 4 |119888|

times radic15 [1 minus tanh2119901 (119909 minus 119888119905)] tanh2119901 (119909 minus 119888119905))

times (5 minus 8tanh2119901 (119909 minus 119888119905))minus1

) + 119888

(42)or

119906 (119909 119905) = (( minus 20119888 [1 minus tanh2119901 (119909 minus 119888119905)] minus 4 |119888|

times radic15 [1 minus tanh2119901 (119909 minus 119888119905)] tanh2119901 (119909 minus 119888119905))

times (5 minus 8 tanh2119901 (119909 minus 119888119905))minus1

) + 119888

(43)

where 119901(119909 minus 119888119905) = radic119886|119888|(|119909 minus 119888119905| minus 119903)

International Journal of Analysis 5

4 Numerical Simulation Examples

Example 1 In (12) take 119888 = 1 ℎ = 0 and 119886 = 3 Then thesolitary wave solution is 119906(119909 119905) = 119890minus|119909minus119905|

We want to show figures with peakon feature But to savespace omitting figures we only give MATLAB program ldquo119909 =minus10 0001 10 119906 = exp(minusabs(119909)) plot(119909 119906)rdquo for 119905 = 0 andldquofor 119899 = 1 101 119909(119899) = (119899 minus 1)5 minus 10 119905(119899) = 119909(119899) end[119909(119899) 119905(119899)] = meshgrid(119909(119899) 119905(119899)) for 119899 = 1 101 for 119898 =

1 101 119906(119899119898) = exp(minusabs(119909(119899) minus 119905(119898))) end end mesh(119906)rdquofor the 3-dimensional case

Example 2 In (16) take 119888 = 1 ℎ = 0 and 119886 = 24 Thensolitary wave solution is 119906(119909 119905) = tanh2(|119909 minus 119905| minus 1) minus 1

The figures with peakon feature will be constructed byldquo119909 = minus10 0001 10 119906 = (tanh(abs(119909)minus1)) and2minus1 plot(119909 119906)rdquofor 119905 = 0 and ldquofor 119899 = 1 101 119909(119899) = (119899minus1)5minus10 119905(119899) = 119909(119899)end [119909(119899) 119905(119899)] = meshgrid(119909(119899) 119905(119899)) for 119899 = 1 101 for119898 = 1 101 119906(119899119898) = (tanh(abs(119909(119899) minus 119905(119898)) minus 1)) and2 minus 1end end mesh(119906)rdquo for the 3-dimensional case

Example 3 In (20) take 119888 = 1 ℎ = 1 and 119886 = 10 Thensymmetrically solitary wave solution is

119906 (119909 119905)

= minus 2 [1 minus tanh2 (|119909 minus 119905| minus 1)]

+2radic3 [1 minus tanh2 (|119909 minus 119905| minus 1)] tanh2 (|119909 minus 119905| minus 1)

times (1 minus 4tanh2 (|119909 minus 119905| minus 1))minus1

(44)

For 119905 = 0 when (119886 = minus2 119887 = 2) (119886 = 231 119887 = 232) and(119886 = 232 119887 = 5) respectively the figures are with peakonfeature 119909 = a 00001 b119910 = (minus2lowast(1minustanh(abs(119909)minus1) and2)+2lowastsqrt(3lowast(1minustanh(abs(119909)minus1) and2)lowasttanh(abs(119909))))(1minus4lowast(1minustanh(abs(119909)minus1) and2)) plot(119909 119910) Replace 119909 by 119909(119899)minus119905(119898)and the others are similar to Example 2 program for the 3-dimensional case

5 Conclusions

In this paper we make simplification of the nonlineargeneralized Camassa-Holm equation and give the improvedansatz for the generalizedCamassa-Holm equationThe threenumerical simulation examples with mATLAB programsdemonstrate solitary wave solutions with peakon featurewhich show our method applicable This method may beapplied to many other nonlinear equations

Acknowledgments

This work was supported partially by National NaturalScience Foundation of China (Grant nos 10871056 and10971150) and by Science Research Foundation in HarbinInstitute of Technology (Grant no HITC200708)

References

[1] R Camassa and D D Holm ldquoAn integrable shallow waterequation with peaked solitonsrdquo Physical Review Letters vol 71no 11 pp 1661ndash1664 1993

[2] Y Long Z Li and W Rui ldquoNew travelling wave solutionsfor a nonlinearly dispersive wave equation of Camassa-Holmequation typerdquo Applied Mathematics and Computation vol 217no 4 pp 1315ndash1320 2010

[3] Y Xu and L Tian ldquoAttractor for a coupled nonhomogeneousCamassa-Holm equationrdquo International Journal of NonlinearScience vol 9 no 1 pp 118ndash122 2010

[4] M Song and Z R Liu ldquoTraveling wave solutions for thegeneralized Zakharov equationsrdquo Mathematical Problems inEngineering vol 2012 Article ID 747295 14 pages 2012

[5] S Tang Y Xiao and Z Wang ldquoTravelling wave solutionsfor a class of nonlinear fourth order variant of a generalizedCamassa-Holm equationrdquo Applied Mathematics and Computa-tion vol 210 no 1 pp 39ndash47 2009

[6] B He W Rui C Chen and S Li ldquoExact travelling wavesolutions of a generalized Camassa-Holm equation using theintegral bifurcationmethodrdquoAppliedMathematics and Compu-tation vol 206 no 1 pp 141ndash149 2008

[7] A-M Wazwaz ldquoNew solitary wave solutions to the modifiedforms of Degasperis-Procesi and Camassa-Holm equationsrdquoApplied Mathematics and Computation vol 186 no 1 pp 130ndash141 2007

[8] A-M Wazwaz ldquoPeakons kinks compactons and solitarypatterns solutions for a family of Camassa-Holm equationsby using new hyperbolic schemesrdquo Applied Mathematics andComputation vol 182 no 1 pp 412ndash424 2006

[9] S Chen C Foias D DHolm E Olson E S Titi and SWynneldquoCamassa-Holm equations as a closure model for turbulentchannel and pipe flowrdquo Physical Review Letters vol 81 no 24pp 5338ndash5341 1998

[10] F Cooper and H Shepard ldquoSolitons in the Camassa-Holmshallow water equationrdquo Physics Letters A vol 194 no 4 pp246ndash250 1994

[11] M Fisher and J Schiff ldquoThe Camassa Holm equation con-served quantities and the initial value problemrdquo Physics LettersA vol 259 no 5 pp 371ndash376 1999

[12] Z Liu T Qian and M Tang ldquoPeakons of the Camassa-Holmequationrdquo Applied Mathematical Modelling vol 26 pp 473ndash480 2002

[13] Z Liu and T Qian ldquoPeakons and their bifurcation in ageneralized Camassa-Holm equationrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol11 no 3 pp 781ndash792 2001

[14] T Qian and M Tang ldquoPeakons and periodic cusp waves ina generalized Camassa-Holm equationrdquo Chaos Solitons andFractals vol 12 no 7 pp 1347ndash1360 2001

[15] R A Kraenkel M Senthilvelan and A I Zenchuk ldquoOnthe integrable perturbations of the Camassa-Holm equationrdquoJournal of Mathematical Physics vol 41 no 5 pp 3160ndash31692000

[16] L Tian and J Yin ldquoNew compacton solutions and solitarywave solutions of fully nonlinear generalized Camassa-Holmequationsrdquo Chaos Solitons and Fractals vol 20 no 2 pp 289ndash299 2004

6 International Journal of Analysis

[17] Z Liu T Jiang P Qin and Q Xu ldquoTrigonometric functionperiodic wave solutions and their limit forms for the KdV-like and the PC-like equationsrdquo Mathematical Problems inEngineering vol 2011 Article ID 810217 23 pages 2011

[18] S A Khuri ldquoNew ansatz for obtaining wave solutions ofthe generalized Camassa-Holm equationrdquo Chaos Solitons ampFractals vol 25 no 3 pp 705ndash710 2005

[19] L Tian and X Song ldquoNew peaked solitary wave solutions ofthe generalized Camassa-Holm equationrdquo Chaos Solitons andFractals vol 19 no 3 pp 621ndash637 2004

[20] R Camassa andA I Zenchuk ldquoOn the initial value problem fora completely integrable shallow water wave equationrdquo PhysicsLetters A vol 281 no 1 pp 26ndash33 2001

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 International Journal of Analysis

where 119903 is an arbitrary constant So

radic3

119886

ln[

[

(120579 +

3 (2119896 minus 119888 + 4119886119888)

8119886

)

+radic1205792+

3 (2119896 minus 119888 + 4119886119888)

4119886

120579+

3 (2119896119888 minus 1198882+ 21198861198882+ 119887)

2119886

]

]

= 119903 minus10038161003816100381610038161205851003816100381610038161003816

(28)

Solving (28) we have that when 3(2119896119888 minus 1198882 + 21198861198882 + 119887)(2119886) minus9(2119896 minus 119888 + 4119886119888)

2(161198862) gt 0

120579 =radic3 (2119896119888 minus 119888

2+ 21198861198882+ 119887)

2119886

minus

9(2119896 minus 119888 + 4119886119888)2

161198862

times sinh[radic3119886

(119903 minus10038161003816100381610038161205851003816100381610038161003816)] minus

3 (2119896 minus 119888 + 4119886119888)

8119886

(29)

when 3(2119896119888minus1198882+21198861198882+119887)(2119886)minus9(2119896minus119888+4119886119888)2(161198862) lt 0

120579 =radic3 (2119896119888 minus 119888

2+ 21198861198882+ 119887)

2119886

minus

9(2119896 minus 119888 + 4119886119888)2

161198862

times cosh[radic3119886

(119903 minus10038161003816100381610038161205851003816100381610038161003816)] minus

3 (2119896 minus 119888 + 4119886119888)

8119886

(30)

Equations (29) and (30) have no asymptotic behavior When3(2119896119888minus119888

2+21198861198882+119887)(2119886) = 9(2119896minus119888+4119886119888)

2(161198862) we obtain

120579 =

1

2

eradic(1198863)(119903minus|120585|) minus 3 (2119896 minus 119888 + 4119886119888)16119886

(31)

So

119906 (119909 119905) =

1

2

eradic(1198863)(119903minus|119909minus119888119905|) minus 6119896 minus 3119888 minus 411988611988816119886

(32)

These solitary solutions have the first derivative discontinuityat the wave peak

When119898 = 2 let 119892 = 0 Consider the following

int(

119886

6

1205793+

2119886119888

3

1205792+

1

4

(2119896 minus 119888 + 41198861198882) 120579

+

1

2

(2119896 minus 1198882+

41198861198883

3

+ 119887))

minus12

d120579 = 119903 minus 10038161003816100381610038161205851003816100381610038161003816

(33)

Take 2119896 minus 119888 + 41198861198882 = 0 and 2119896 minus 1198882 + 411988611988833 + 119887 = 0 then

minusradic6

119886119888

atanhradic120579 + 41198884119888

= 119903 minus10038161003816100381610038161205851003816100381610038161003816 (34)

From (34) we have

120579 = 4119888 tanh2 [radic1198861198886

(10038161003816100381610038161205851003816100381610038161003816minus 119903)] minus 4119888 (35)

Therefore we have the solitary wave solutions of the general-ized Camassa-Holm equation as

119906 (119909 119905) = 4119888 tanh2 [radic1198861198886

(|119909 minus 119888119905| minus 119903)] minus 3119888 (36)

Take 2119896minus119888+41198861198882 = 811988611988823 and 2119896minus1198882 +411988611988833+119887 = 0 then

radic12

119886119888

atanradic 1205792119888

= 119903 minus10038161003816100381610038161205851003816100381610038161003816 (37)

Similarly we have

119906 (119909 119905) = 2119888 tan2 [radic 11988611988812

(119903 minus |119909 minus 119888119905|)] + 119888 (38)

This result is the same as (28) in [18] But it is easily to obtainWhen119898 = 3 let 119892 = 0 Consider the following

int(

119886

10

1205794+

119886119888

2

1205793+ 11988611988821205792+

1

4

(2119896 minus 119888 + 41198861198883) 120579

+

1

2

(2119896 minus 1198882+ 1198861198884+ 119887))

minus12

d120579 = 119903 minus 10038161003816100381610038161205851003816100381610038161003816

(39)

From (39) and 119886 119888 gt 0 we have seen that 119865(d119865d120585) is not apolynomial in 119865 (compared with Equation (35) in [18]) Take2119896 minus 119888 + 4119886119888

3= 0 and 2119896 minus 1198882 + 1198861198884 + 119887 = 0 then

minusradic1

1198861198882atanh

radic10 (119888120579 + 41198882)

4 |119888| radic1205792+ 5119888120579 + 10119888

2

= 119903 minus10038161003816100381610038161205851003816100381610038161003816 (40)

Solving (40) we obtain

120579 = ( minus 20119888 [1 minus tanh2119901 (120585)]

plusmn 4 |119888| radic15 [1 minus tanh2119901 (120585)] tanh2119901 (120585))

times (5 minus 8tanh2119901 (120585))minus1

(41)

where 119901(120585) = radic119886|119888|(|120585| minus 119903) Therefore

119906 (119909 119905) = (( minus 20119888 [1 minus tanh2119901 (119909 minus 119888119905)] + 4 |119888|

times radic15 [1 minus tanh2119901 (119909 minus 119888119905)] tanh2119901 (119909 minus 119888119905))

times (5 minus 8tanh2119901 (119909 minus 119888119905))minus1

) + 119888

(42)or

119906 (119909 119905) = (( minus 20119888 [1 minus tanh2119901 (119909 minus 119888119905)] minus 4 |119888|

times radic15 [1 minus tanh2119901 (119909 minus 119888119905)] tanh2119901 (119909 minus 119888119905))

times (5 minus 8 tanh2119901 (119909 minus 119888119905))minus1

) + 119888

(43)

where 119901(119909 minus 119888119905) = radic119886|119888|(|119909 minus 119888119905| minus 119903)

International Journal of Analysis 5

4 Numerical Simulation Examples

Example 1 In (12) take 119888 = 1 ℎ = 0 and 119886 = 3 Then thesolitary wave solution is 119906(119909 119905) = 119890minus|119909minus119905|

We want to show figures with peakon feature But to savespace omitting figures we only give MATLAB program ldquo119909 =minus10 0001 10 119906 = exp(minusabs(119909)) plot(119909 119906)rdquo for 119905 = 0 andldquofor 119899 = 1 101 119909(119899) = (119899 minus 1)5 minus 10 119905(119899) = 119909(119899) end[119909(119899) 119905(119899)] = meshgrid(119909(119899) 119905(119899)) for 119899 = 1 101 for 119898 =

1 101 119906(119899119898) = exp(minusabs(119909(119899) minus 119905(119898))) end end mesh(119906)rdquofor the 3-dimensional case

Example 2 In (16) take 119888 = 1 ℎ = 0 and 119886 = 24 Thensolitary wave solution is 119906(119909 119905) = tanh2(|119909 minus 119905| minus 1) minus 1

The figures with peakon feature will be constructed byldquo119909 = minus10 0001 10 119906 = (tanh(abs(119909)minus1)) and2minus1 plot(119909 119906)rdquofor 119905 = 0 and ldquofor 119899 = 1 101 119909(119899) = (119899minus1)5minus10 119905(119899) = 119909(119899)end [119909(119899) 119905(119899)] = meshgrid(119909(119899) 119905(119899)) for 119899 = 1 101 for119898 = 1 101 119906(119899119898) = (tanh(abs(119909(119899) minus 119905(119898)) minus 1)) and2 minus 1end end mesh(119906)rdquo for the 3-dimensional case

Example 3 In (20) take 119888 = 1 ℎ = 1 and 119886 = 10 Thensymmetrically solitary wave solution is

119906 (119909 119905)

= minus 2 [1 minus tanh2 (|119909 minus 119905| minus 1)]

+2radic3 [1 minus tanh2 (|119909 minus 119905| minus 1)] tanh2 (|119909 minus 119905| minus 1)

times (1 minus 4tanh2 (|119909 minus 119905| minus 1))minus1

(44)

For 119905 = 0 when (119886 = minus2 119887 = 2) (119886 = 231 119887 = 232) and(119886 = 232 119887 = 5) respectively the figures are with peakonfeature 119909 = a 00001 b119910 = (minus2lowast(1minustanh(abs(119909)minus1) and2)+2lowastsqrt(3lowast(1minustanh(abs(119909)minus1) and2)lowasttanh(abs(119909))))(1minus4lowast(1minustanh(abs(119909)minus1) and2)) plot(119909 119910) Replace 119909 by 119909(119899)minus119905(119898)and the others are similar to Example 2 program for the 3-dimensional case

5 Conclusions

In this paper we make simplification of the nonlineargeneralized Camassa-Holm equation and give the improvedansatz for the generalizedCamassa-Holm equationThe threenumerical simulation examples with mATLAB programsdemonstrate solitary wave solutions with peakon featurewhich show our method applicable This method may beapplied to many other nonlinear equations

Acknowledgments

This work was supported partially by National NaturalScience Foundation of China (Grant nos 10871056 and10971150) and by Science Research Foundation in HarbinInstitute of Technology (Grant no HITC200708)

References

[1] R Camassa and D D Holm ldquoAn integrable shallow waterequation with peaked solitonsrdquo Physical Review Letters vol 71no 11 pp 1661ndash1664 1993

[2] Y Long Z Li and W Rui ldquoNew travelling wave solutionsfor a nonlinearly dispersive wave equation of Camassa-Holmequation typerdquo Applied Mathematics and Computation vol 217no 4 pp 1315ndash1320 2010

[3] Y Xu and L Tian ldquoAttractor for a coupled nonhomogeneousCamassa-Holm equationrdquo International Journal of NonlinearScience vol 9 no 1 pp 118ndash122 2010

[4] M Song and Z R Liu ldquoTraveling wave solutions for thegeneralized Zakharov equationsrdquo Mathematical Problems inEngineering vol 2012 Article ID 747295 14 pages 2012

[5] S Tang Y Xiao and Z Wang ldquoTravelling wave solutionsfor a class of nonlinear fourth order variant of a generalizedCamassa-Holm equationrdquo Applied Mathematics and Computa-tion vol 210 no 1 pp 39ndash47 2009

[6] B He W Rui C Chen and S Li ldquoExact travelling wavesolutions of a generalized Camassa-Holm equation using theintegral bifurcationmethodrdquoAppliedMathematics and Compu-tation vol 206 no 1 pp 141ndash149 2008

[7] A-M Wazwaz ldquoNew solitary wave solutions to the modifiedforms of Degasperis-Procesi and Camassa-Holm equationsrdquoApplied Mathematics and Computation vol 186 no 1 pp 130ndash141 2007

[8] A-M Wazwaz ldquoPeakons kinks compactons and solitarypatterns solutions for a family of Camassa-Holm equationsby using new hyperbolic schemesrdquo Applied Mathematics andComputation vol 182 no 1 pp 412ndash424 2006

[9] S Chen C Foias D DHolm E Olson E S Titi and SWynneldquoCamassa-Holm equations as a closure model for turbulentchannel and pipe flowrdquo Physical Review Letters vol 81 no 24pp 5338ndash5341 1998

[10] F Cooper and H Shepard ldquoSolitons in the Camassa-Holmshallow water equationrdquo Physics Letters A vol 194 no 4 pp246ndash250 1994

[11] M Fisher and J Schiff ldquoThe Camassa Holm equation con-served quantities and the initial value problemrdquo Physics LettersA vol 259 no 5 pp 371ndash376 1999

[12] Z Liu T Qian and M Tang ldquoPeakons of the Camassa-Holmequationrdquo Applied Mathematical Modelling vol 26 pp 473ndash480 2002

[13] Z Liu and T Qian ldquoPeakons and their bifurcation in ageneralized Camassa-Holm equationrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol11 no 3 pp 781ndash792 2001

[14] T Qian and M Tang ldquoPeakons and periodic cusp waves ina generalized Camassa-Holm equationrdquo Chaos Solitons andFractals vol 12 no 7 pp 1347ndash1360 2001

[15] R A Kraenkel M Senthilvelan and A I Zenchuk ldquoOnthe integrable perturbations of the Camassa-Holm equationrdquoJournal of Mathematical Physics vol 41 no 5 pp 3160ndash31692000

[16] L Tian and J Yin ldquoNew compacton solutions and solitarywave solutions of fully nonlinear generalized Camassa-Holmequationsrdquo Chaos Solitons and Fractals vol 20 no 2 pp 289ndash299 2004

6 International Journal of Analysis

[17] Z Liu T Jiang P Qin and Q Xu ldquoTrigonometric functionperiodic wave solutions and their limit forms for the KdV-like and the PC-like equationsrdquo Mathematical Problems inEngineering vol 2011 Article ID 810217 23 pages 2011

[18] S A Khuri ldquoNew ansatz for obtaining wave solutions ofthe generalized Camassa-Holm equationrdquo Chaos Solitons ampFractals vol 25 no 3 pp 705ndash710 2005

[19] L Tian and X Song ldquoNew peaked solitary wave solutions ofthe generalized Camassa-Holm equationrdquo Chaos Solitons andFractals vol 19 no 3 pp 621ndash637 2004

[20] R Camassa andA I Zenchuk ldquoOn the initial value problem fora completely integrable shallow water wave equationrdquo PhysicsLetters A vol 281 no 1 pp 26ndash33 2001

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

International Journal of Analysis 5

4 Numerical Simulation Examples

Example 1 In (12) take 119888 = 1 ℎ = 0 and 119886 = 3 Then thesolitary wave solution is 119906(119909 119905) = 119890minus|119909minus119905|

We want to show figures with peakon feature But to savespace omitting figures we only give MATLAB program ldquo119909 =minus10 0001 10 119906 = exp(minusabs(119909)) plot(119909 119906)rdquo for 119905 = 0 andldquofor 119899 = 1 101 119909(119899) = (119899 minus 1)5 minus 10 119905(119899) = 119909(119899) end[119909(119899) 119905(119899)] = meshgrid(119909(119899) 119905(119899)) for 119899 = 1 101 for 119898 =

1 101 119906(119899119898) = exp(minusabs(119909(119899) minus 119905(119898))) end end mesh(119906)rdquofor the 3-dimensional case

Example 2 In (16) take 119888 = 1 ℎ = 0 and 119886 = 24 Thensolitary wave solution is 119906(119909 119905) = tanh2(|119909 minus 119905| minus 1) minus 1

The figures with peakon feature will be constructed byldquo119909 = minus10 0001 10 119906 = (tanh(abs(119909)minus1)) and2minus1 plot(119909 119906)rdquofor 119905 = 0 and ldquofor 119899 = 1 101 119909(119899) = (119899minus1)5minus10 119905(119899) = 119909(119899)end [119909(119899) 119905(119899)] = meshgrid(119909(119899) 119905(119899)) for 119899 = 1 101 for119898 = 1 101 119906(119899119898) = (tanh(abs(119909(119899) minus 119905(119898)) minus 1)) and2 minus 1end end mesh(119906)rdquo for the 3-dimensional case

Example 3 In (20) take 119888 = 1 ℎ = 1 and 119886 = 10 Thensymmetrically solitary wave solution is

119906 (119909 119905)

= minus 2 [1 minus tanh2 (|119909 minus 119905| minus 1)]

+2radic3 [1 minus tanh2 (|119909 minus 119905| minus 1)] tanh2 (|119909 minus 119905| minus 1)

times (1 minus 4tanh2 (|119909 minus 119905| minus 1))minus1

(44)

For 119905 = 0 when (119886 = minus2 119887 = 2) (119886 = 231 119887 = 232) and(119886 = 232 119887 = 5) respectively the figures are with peakonfeature 119909 = a 00001 b119910 = (minus2lowast(1minustanh(abs(119909)minus1) and2)+2lowastsqrt(3lowast(1minustanh(abs(119909)minus1) and2)lowasttanh(abs(119909))))(1minus4lowast(1minustanh(abs(119909)minus1) and2)) plot(119909 119910) Replace 119909 by 119909(119899)minus119905(119898)and the others are similar to Example 2 program for the 3-dimensional case

5 Conclusions

In this paper we make simplification of the nonlineargeneralized Camassa-Holm equation and give the improvedansatz for the generalizedCamassa-Holm equationThe threenumerical simulation examples with mATLAB programsdemonstrate solitary wave solutions with peakon featurewhich show our method applicable This method may beapplied to many other nonlinear equations

Acknowledgments

This work was supported partially by National NaturalScience Foundation of China (Grant nos 10871056 and10971150) and by Science Research Foundation in HarbinInstitute of Technology (Grant no HITC200708)

References

[1] R Camassa and D D Holm ldquoAn integrable shallow waterequation with peaked solitonsrdquo Physical Review Letters vol 71no 11 pp 1661ndash1664 1993

[2] Y Long Z Li and W Rui ldquoNew travelling wave solutionsfor a nonlinearly dispersive wave equation of Camassa-Holmequation typerdquo Applied Mathematics and Computation vol 217no 4 pp 1315ndash1320 2010

[3] Y Xu and L Tian ldquoAttractor for a coupled nonhomogeneousCamassa-Holm equationrdquo International Journal of NonlinearScience vol 9 no 1 pp 118ndash122 2010

[4] M Song and Z R Liu ldquoTraveling wave solutions for thegeneralized Zakharov equationsrdquo Mathematical Problems inEngineering vol 2012 Article ID 747295 14 pages 2012

[5] S Tang Y Xiao and Z Wang ldquoTravelling wave solutionsfor a class of nonlinear fourth order variant of a generalizedCamassa-Holm equationrdquo Applied Mathematics and Computa-tion vol 210 no 1 pp 39ndash47 2009

[6] B He W Rui C Chen and S Li ldquoExact travelling wavesolutions of a generalized Camassa-Holm equation using theintegral bifurcationmethodrdquoAppliedMathematics and Compu-tation vol 206 no 1 pp 141ndash149 2008

[7] A-M Wazwaz ldquoNew solitary wave solutions to the modifiedforms of Degasperis-Procesi and Camassa-Holm equationsrdquoApplied Mathematics and Computation vol 186 no 1 pp 130ndash141 2007

[8] A-M Wazwaz ldquoPeakons kinks compactons and solitarypatterns solutions for a family of Camassa-Holm equationsby using new hyperbolic schemesrdquo Applied Mathematics andComputation vol 182 no 1 pp 412ndash424 2006

[9] S Chen C Foias D DHolm E Olson E S Titi and SWynneldquoCamassa-Holm equations as a closure model for turbulentchannel and pipe flowrdquo Physical Review Letters vol 81 no 24pp 5338ndash5341 1998

[10] F Cooper and H Shepard ldquoSolitons in the Camassa-Holmshallow water equationrdquo Physics Letters A vol 194 no 4 pp246ndash250 1994

[11] M Fisher and J Schiff ldquoThe Camassa Holm equation con-served quantities and the initial value problemrdquo Physics LettersA vol 259 no 5 pp 371ndash376 1999

[12] Z Liu T Qian and M Tang ldquoPeakons of the Camassa-Holmequationrdquo Applied Mathematical Modelling vol 26 pp 473ndash480 2002

[13] Z Liu and T Qian ldquoPeakons and their bifurcation in ageneralized Camassa-Holm equationrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol11 no 3 pp 781ndash792 2001

[14] T Qian and M Tang ldquoPeakons and periodic cusp waves ina generalized Camassa-Holm equationrdquo Chaos Solitons andFractals vol 12 no 7 pp 1347ndash1360 2001

[15] R A Kraenkel M Senthilvelan and A I Zenchuk ldquoOnthe integrable perturbations of the Camassa-Holm equationrdquoJournal of Mathematical Physics vol 41 no 5 pp 3160ndash31692000

[16] L Tian and J Yin ldquoNew compacton solutions and solitarywave solutions of fully nonlinear generalized Camassa-Holmequationsrdquo Chaos Solitons and Fractals vol 20 no 2 pp 289ndash299 2004

6 International Journal of Analysis

[17] Z Liu T Jiang P Qin and Q Xu ldquoTrigonometric functionperiodic wave solutions and their limit forms for the KdV-like and the PC-like equationsrdquo Mathematical Problems inEngineering vol 2011 Article ID 810217 23 pages 2011

[18] S A Khuri ldquoNew ansatz for obtaining wave solutions ofthe generalized Camassa-Holm equationrdquo Chaos Solitons ampFractals vol 25 no 3 pp 705ndash710 2005

[19] L Tian and X Song ldquoNew peaked solitary wave solutions ofthe generalized Camassa-Holm equationrdquo Chaos Solitons andFractals vol 19 no 3 pp 621ndash637 2004

[20] R Camassa andA I Zenchuk ldquoOn the initial value problem fora completely integrable shallow water wave equationrdquo PhysicsLetters A vol 281 no 1 pp 26ndash33 2001

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 International Journal of Analysis

[17] Z Liu T Jiang P Qin and Q Xu ldquoTrigonometric functionperiodic wave solutions and their limit forms for the KdV-like and the PC-like equationsrdquo Mathematical Problems inEngineering vol 2011 Article ID 810217 23 pages 2011

[18] S A Khuri ldquoNew ansatz for obtaining wave solutions ofthe generalized Camassa-Holm equationrdquo Chaos Solitons ampFractals vol 25 no 3 pp 705ndash710 2005

[19] L Tian and X Song ldquoNew peaked solitary wave solutions ofthe generalized Camassa-Holm equationrdquo Chaos Solitons andFractals vol 19 no 3 pp 621ndash637 2004

[20] R Camassa andA I Zenchuk ldquoOn the initial value problem fora completely integrable shallow water wave equationrdquo PhysicsLetters A vol 281 no 1 pp 26ndash33 2001

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