(G′/G2)-expansion Solutions to MBBM and OBBM Equations

KANG Zhouzheng

College of Mathematics,Inner Mongolia University for Nationalities, Tongliao 028043,China.

(G′/G2)-expansion Solutions to MBBM and OBBM Equations

KANG Zhouzheng?

College of Mathematics,Inner Mongolia University for Nationalities, Tongliao 028043,China.

.In this paper,with the aid of symbolic computation,the modified Benjamin-Bona-Mahony and Ostrovsky-Benjamin-Bona-Mahony equations are investigated by extended(G′/G2)-expansion method.As a consequence,some trigonometric,hyperbolic and rational function solutions with multiple arbitrary parameters for the two equations are revealed,which helps to illustrate the effectiveness of this method.

Modified Benjamin-Bona-Mahony equation;Ostrovsky-Benjamin-Bona-Mahony equation;exact solutions.

1 Introduction

Because of the profound significance of nonlinear evolution equations(NLEEs)arising from physics and applied sciences,researchers have devoted a lot of effort to study them. One of the fundamental problems is to construct as many and general as possible exact solutions of NLEEs.Rather surprisingly,there are a number of methods have been well established.One of these methods is the basic(G′/G)-expansion method[1],which has been widely utilized to seek many exact solutions of NLEEs[2,3],discrete nonlinear equations[4],and fractional differential equations[5].

Based on the basic(G′/G)-expansion method,many advances have been made in different manners such as the extended(G′/G)-expansion method[6],the generalized (G′/G)-expansion method[7,8],the modified(G′/G)-expansion method[9],the improved(G′/G)-expansion method[10],the two-variable(G′/G,1/G)-expansion method[11],the extended discrete(G′/G)-expansion method[12],the generalized and improved (G′/G)-expansion method[13],the further improved(G′/G)-expansion method[14], the multiple(G′/G)-expansion method[15],the generalized extended(G′/G)-expansion method[16],the(ω/g)-expansion method[17],and so on.These developed methods are used to find traveling wave solutions of many differential equations.

This article is arranged as follows.In Section 2,we outline the main steps of the extended method.In Section 3,we apply the extended method to the modified Benjamin-Bona-Mahony(MBBM)and Ostrovsky-Benjamin-Bona-Mahony(OBBM)equations respectively to find out many exact solutions.Lastly,a brief summary of the results is given.

2 Methodology

Consider a given NLEE withu=u(t,x,y,···)as

Step 1.Reduction of a NLEE(2.1)into an ODE(2.2).Introducing a variationu(t,x,y, ···)=u(ξ),ξ=sx+ly+···?Vt+d,wheres,l,···,V,dare undetermined constants,we can convert Eq.(2.1)into an ODE

Step 2.Construction of the formal solution of Eq.(2.2).We assume that Eq.(2.2)has the solution in the form

whereG=G(ξ)satisfies

in whichμ/=1 andλ/=0 are integers,whilea0,ai,bi(i=1,2,···,m)are constants to be determined.From Eq.(2.2),we can determine the positive integermby homogeneous balance principle.

Step 3.Determination of unknown constants.Substituting(2.3)along with(2.4)into Eq.(2.2),collecting all terms with the same powers of(G′/G2)j(j=0,±1,±2,···), and equating zero of all the coefficients yields a set of algebraic equations.By simple calculation with symbolic computation software,the solutions of the algebraic equations can be derived.

Step 4.Explicit expression of exact solutions to Eq.(2.1).According to the general solutions of Eq.(2.4),we have the following results.

Ifμλ＞0,then we can get the ratio

Ifμλ＜0,then we can get the ratio

Ifμ=0 andλ/=0,then we can get the ratio

whereAandBare arbitrary nonzero constants.By substituting the values ofa0,ai,bi(i= 1,2,···,m),s,l,···,V,dand(2.5)-(2.7)into(2.3),we can obtain the solutions of Eq.(2.1).

3 Applications

Next,via the above method,we would like to consider two NLEEs.

Example 1.The(1+1)-dimensional MBBM equation[18]takes the form

This equation is used to model an approximation for surface long waves in nonlinear dispersive media,and also characterize the hydromagnetic waves in cold plasma,acoustic waves in anharmonic crystals and acoustic gravity waves in compressible fluids.

Under the transformation

Eq.(3.1)is converted into an ODE with respect toξ,namely,

Analysis of Eq.(3.3)with the homogeneous balance principle reveals that the formal solution can be written asin whicha0,a1,b1are undetermined constants.Substituting(3.4)along with(2.4)into Eq. (3.3),and collecting all terms with the same powers of(G′/G2)j(j=0,±1,±2,±3),we set all the coefficients to zero,which leads to a set of algebraic equations.Here we omit to display them for simplicity.Solving the resulting equations with Maple,we can derive

According to Cases 1-2 and(2.5)-(2.7),therefore,exact solutions of Eq.(3.1)can be given below

Family 1.Whenμλ＞0,we can get the trigonometric function solutions of Eq.(3.1)

Whenμλ＜0,we can get the hyperbolic function solutions of Eq.(3.1)

Whenμ=0 andλ0,we can get the rational function solutions of Eq.(3.1)

Family 2.Whenμλ＞0,we can get the trigonometric function solutions of Eq.(3.1)

Whenμλ＜0,we can get the hyperbolic function solutions of Eq.(3.1)

Whenμ=0 andλ0,we can get the rational function solutions of Eq.(3.1)

whereξ=x?(8μλ+1)t.

Example 2.The(1+1)-dimensional OBBM equation reads

which describes the motion of ocean currents.Whenγ=0,integrating once with respect tox,and letting the integration constant be zero,Eq.(3.5)becomes the well known Benjamin-Bona-Mahony equation.In[19],traveling wave solutions of Eq.(3.5)are constructed by the simplified(G′/G)-expansion method.

Through the transformation(3.2),Eq.(3.5)becomes

By balancing the highest orderderivative with the nonlinear terms in Eq.(3.6),the formal solution can be expressed as

Performing similar calculation as Example 1 gives

From the Cases 1-6 and(2.5)-(2.7),the solutions of Eq.(3.5)we are looking for can be listed as follows

Family 1.WhenEq.(3.5)possesses the trigonometric function solutions

Family 2.When(3.5)possesses the trigonometric function solutions

Family 3.WhenEq.(3.5)possesses the trigonometric function solutions

Family 4.When(3.5)possesses the trigonometric function solutions

Family 5.When(3.5)possesses the trigonometric function solutions

Family 6.When(3.5)possesses the trigonometric function solutions

4 Conclusion

In summary,we have successfully obtained abundant solutions of the MBBM and OBBM equations using the extended(G′/G2)-expansion method and symbolic computation. The obtained solutions are expressed in terms of trigonometric,hyperbolic and rational functions,and include several arbitrary parameters.An advantage of this method is that it is applicable to a number of NLEEs in mathematical physics.

Acknowledgments

This work is supported by the NSFC(11462019)and the Scientific Research Foundation of Inner Mongolia University for Nationalities(NMD1306).The author would like to thank the referees for helpful comments and suggestions.

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Received 16 March 2015;Accepted 19 April 2015

?Corresponding author.Email address:zhzhkang@126.com(Z.Z.Kang)

AMS Subject Classifications:35Q53,35C07

Chinese Library Classifications:O175.29