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Open Physics

formerly Central European Journal of Physics

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Volume 16, Issue 1


Volume 13 (2015)

Exact solutions and conservation laws for the modified equal width-Burgers equation

Chaudry Masood Khalique
  • Corresponding author
  • International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences,North-West University Mmabatho Mafikeng Campus, Private Bag X 2046, Mmabatho 2735, Republic of South Africa
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  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Innocent Simbanefayi
  • International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences,North-West University Mmabatho Mafikeng Campus, Private Bag X 2046, Mmabatho 2735, Republic of South Africa
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  • Other articles by this author:
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Published Online: 2018-12-26 | DOI: https://doi.org/10.1515/phys-2018-0099


In this paper we study the modified equal width-Burgers equation, which describes long wave propagation in nonlinear media with dispersion and dissipation. Using the Lie symmetry method in conjunction with the (G'/G)− expansion method we construct its travelling wave solutions. Also, we determine the conservation laws by invoking the new conservation theorem due to Ibragimov. As a result we obtain energy and linear momentum conservation laws.

Keywords: Modified equal width-Burgers equation; Lie point symmetries; exact solutions; conservation laws

PACS: 02.30.Jr; 04.20.Jb; 11.30.-j

1 Introduction

It is well-known that majority of the real-world physical phenomena are modeled by mathematical equations, especially nonlinear partial differential equations (NLPDEs). These phenomena include the problems from fluid mechanics, elasticity, plasma and optical fibers, relativity, gas dynamics, thermodynamics, and many more. In order to comprehend the understanding of such physical phenomena it is vital to look for exact solutions of the NLPDEs. During the last sixty years many scientists and mathematicians developed several effective and useful methods for obtaining exact solutions of NLPDEs. These include the simplified Hirota’s method [1, 2], the tanh-coth method [3], the sine-cosine method [4], the simplest equation method[5], the homogeneous balance method [6], the inverse scattering transform method [7], Hirota’s bilinear method [8], the (G'/G)−expansion method [9, 10], Riccati-Bernoulli sub-ordinary differential equation method [11], Jacobi elliptic function expansion method[12, 13], Kudryashov method [14, 15], the Lie symmetry method [16, 17, 18, 19, 20, 21], just to mention a few.

On the other hand conservation laws are very important in the study of NLPDEs and much research has been done on different methods of obtaining conservation laws. Conservation laws are essential in determining the extent of integrability of differential equations, development of numerical schemes, reduction and solutions of partial differential equations. See, for example [22, 23, 24, 25, 26, 27, 28, 29] and references therein.

The Burgers equation


where v is a constant defining the kinematic viscosity models the turbulent flow in a channel and describes the effect of coupling between diffusion and convection processes on a fluid. It first appeared in academic circles in [30]. Over the years, various researchers have applied modifications to (1.1) and used many different methods to study the equation. For example, the equal width equation


was first introduced in [31]. It describes amongst others, nonlinear dispersive waves such as shallow water waves and nonlinear waves in plasmas. Extensive work was done on Equation (1.2) in constructing numerical solutions [32, 33]. Arora et al. [34] performed the reduced differential transform method to find the numerical solution of the equal width wave equation and the exact analytical solution of the inviscid Burgers equation with initial conditions.

The modified equal width equation


was studied in [35] and solitary wave solutions were obtained. Furthermore, using Quintic B-spline method the interactions through computer simulation were observed. In [36] the sine-cosine and the tanh methods were employed to obtain exact solutions of (1.3) and two of its variants, which included compactons, solitons, solitary patterns, and periodic solutions. Hasan [37] presented the numerical solution for (1.3) using Fourier spectral method that discretizes the space variable and Leap-frog method scheme for time dependence.

The generalised equal width equation


was studied in [38] and its solitary wave solutions were obtained by a collocation method using quadratic B-spline at the midpoints.

In [39] the author introduced the modified equal width-Burgers (MEW-Burgers) equation


which describes long wave propagation in nonlinear media with dispersion and dissipation. Here α, β are positive parameters and ω is a damping parameter. The bifurcation behaviour and an external periodic perturbation of the MEW-Burgers equation (1.5) was studied [39].

In this paper using the Lie symmetry method and the (G0/G)−expansion method we construct travelling wave solutions of (1.5). Furthermore, we derive the conservation laws by applying the new conservation theorem due to Ibragimov.

2 Travelling wave solutions of (1.5)

In this section we obtain travelling wave solutions of (1.5) by employing Lie symmetry analysis together with the (G'/G)−expansion method.

2.1 Lie point symmetries and reduction of (1.5)

We begin by determining the Lie point symmetries of (1.5). The vector field


is a Lie point symmetry of (1.5) if and only if




is the third prolongation of X [18]. Expanding (2.7), we obtain


Splitting the above equation with respect to derivatives of u yields the system of ten overdetermined linear partial differential equations


Solving (2.9)–(2.18) for ξ1, ξ2 and η we obtain


which results in two translation symmetries


A linear combination of these Lie point symmetries, that is, X = X1 + cX2 produces the associated Lagrange equations


which upon solving yields the two invariants z = xct and U = u, and hence the group-invariant solution u = U(z). Taking U and z as the new dependent and independent variables respectively, equation (1.5) is transformed into the third-order nonlinear ordinary differential equation (ODE)


2.2 Solution of (2.20) using (G'/G)−expansion method

In this subsection we use the (G'/G)−expansion method [9] to obtain solutions of (2.20). Firstly, we assume that


is the solution to (2.20), where A0, A1, · · · Am are to be determined. The balancing procedure is used to find the value of m, a positive integer. In our case the balancing procedure yields m = 1, thus (2.21) becomes


Secondly, we substitute (2.22) into (2.20) and simultaneously use the second-order ODE


where λ and μ are arbitrary constants. This yields


Collecting terms with like powers of (G'/G) and equating them to zero gives the overdetermined system of five algebraic equations


Using Mathematica, two solutions of the above system of algebraic equations are

Solution set 1


Solution set 2


Thus corresponding to solution set 1 above, we have the following three types of solutions for the MEW-Burgers (1.5):

Case 1.1 For M = λ2 − 4μ > 0, we obtain the hyperbolic function solution


where Δ1=M/2,z=x2(3μλ2)t/(3λ)and A and B are constants.

Case 1.2 For M = λ2 − 4μ < 0, we obtain the trigonometric function solution


where Δ2=M/2,z=x2(3μλ2)t/(3λ)and A and B are constants.

Case 1.3 For M = λ2 − 4μ = 0, we obtain the rational function solution


where z = x − 2(3μλ2)t/(3λ) and A and B are constants.

Similarly, considering the solution set 2 we obtain the following three types of solutions for the MEW-Burgers (1.5):

Case 2.1 For M = λ2 − 4μ > 0, we obtain the hyperbolic function solution


where Δ1=M/2,z=x+4μωt/3and A and B are constants.

Case 2.2 For M = λ2 − 4μ < 0, we obtain the trigonometric function solution


where Δ2=M/2,z=x+4μωt/3and A and B are constants.

Case 2.3 For M = λ2 − 4μ = 0, we obtain the rational function solution


where z=x+4μωt/3and A and B are constants.

The solution profile of (2.24) for λ = −0.09, A = 0, B = 1, μ = 0.01, ω = 0.03, α = 1 is presented in Figure 1, whereas the solution profile of (2.26) for λ = −0.8, A = 0.4, B = 3, μ = 0.2, ω = 0.01, α = 0.1 is given in Figure 2.

Profile of solution (2.24)
Figure 1

Profile of solution (2.24)

Profile of solution (2.26)
Figure 2

Profile of solution (2.26)

3 Conservation laws of (1.5)

In this section we derive conservation laws for the MEW-Burgers equation (1.5) using the new conservation theorem due to Ibragimov [40, 41].

We begin by determining the adjoint equation of (1.5), namely


using the formula


where δ/u is the Euler-Lagrange operator defined by


with the total differential operators Dt and D x given by


Thus Equation (3.27) becomes


The MEW-Burgers equation (1.5) together with its adjoint equation (3.29) have the Lagrangian


which is equivalent to the second-order Lagrangian


We recall that the MEW-Burgers equation (1.5) admits two translation symmetries X1 = /∂t and X2 = /∂x. To obtain the conserved vectors corresponding to these two infinitesimal generators we use [40]


where W1 and W2 are the Lie characteristic functions. Let us first consider the infinitesimal generator X1 = / ∂t. It can be easily shown that the prolongation of the generator X1 to the derivatives involved in the MEW-Burgers equation (1.5) has the form / ∂t. In order to determine the value

of λ we use equation X1[3](F)=λ(F),which yields λ = 0. Since Dt(ξ1) = 0, we obtain η* = 0 and hence the operator admitted by the adjoint equation (3.29) is


We now use (3.34) to compute the Lie characteristic functions W1 and W2, which in this case are W1 = −ut and W2 = −vt. Thus by using (3.32) and (3.33) the conserved vector for the system (1.5) and (3.29) corresponding to X1 is


Similarly, we compute the conserved vector corresponding to X2. In this case W1 = −ux and W2 = −vx and the conserved vector thus rendered is


Remark: It should be noted that the time translation symmetry gives us the energy conservation law whereas the space translation symmetry provides us with the linear momentum conservation law.

4 Conclusion

In this paper we obtained travelling wave solutions of the MEW-Burgers equation (1.5) The two translation symmetries were used to reduce the MEW-Burgers equation to an ordinary differential equation. The (G'/G)−expansion method was applied to the ordinary differential equation to obtain its solutions. Consequently travelling wave solutions were obtained for the MEW-Burgers equation. These solutions were hyperbolic, trigonometric and rational functions. Furthermore we derived two conservation laws using the new conservation theorem due to Ibragimov. These were the energy and the linear momentum conservation laws.


  • [1]

    Wazwaz A.M., El-Tantawy S.A., A new integrable (3 + 1)- dimensional KdV-like model with its multiple-soliton solutions, Nonlin. Dyn., 2016, 83, 1529-1534. CrossrefGoogle Scholar

  • [2]

    Wazwaz A.M., El-Tantawy S.A., Solving the (3+1)-dimensional KP-Boussinesq and BKP-Boussinesq equations by the simplified Hirota’s method, Nonlin. Dyn., 2017, 88, 3017-3021. CrossrefGoogle Scholar

  • [3]

    Wazwaz A.M., The tanh-coth method for solitons and kink solutions for nonlinear parabolic equations, Appl. Math. Comp., 2007, 188 , 1467-1475. CrossrefGoogle Scholar

  • [4]

    Wazwaz A.M., Exact solutions for the ZK-MEW equation by using the tanh and sine-cosine methods, J. Comp. Math., 2005, 82, 699-708. Google Scholar

  • [5]

    Kudryashov N.A., Exact solitary waves of the Fisher equation, Phys. Lett. A., 2005, 342, 99-106. CrossrefGoogle Scholar

  • [6]

    Wang M., Zhou Y., Li Z., Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics, Phys. Lett. A, 1996, 216, 67-75. CrossrefGoogle Scholar

  • [7]

    Ablowitz M.J., Clarkson P.A., Solitons, Nonlinear evolution equations and inverse scattering, 1991, Cambridge University Press, Cambridge. Google Scholar

  • [8]

    Hirota R., The direct method in soliton theory, 2004, Cambridge University Press, Cambridge. Google Scholar

  • [9]

    Wang M., Li X., Zhang J., The G'/G− expansion method and travellingwave solutions for linear evolution equations inmathematical physics, Phys. Lett. A, 2005, 24, 1257-1268. Google Scholar

  • [10]

    Mhlanga I.E., Khalique C.M., A study of a generalized Benney-Luke equation with time-dependent coefficients, Nonlin. Dyn., 2017, 90, 1535-1544. CrossrefGoogle Scholar

  • [11]

    Baleanu D., Inc M., Yusuf A., Aliyu A.I., Traveling wave solutions and conservation laws for nonlinear evolution equation, J.Math. Phys., 2018, 59, 023506. CrossrefWeb of ScienceGoogle Scholar

  • [12]

    Zhang Z., Jacobi elliptic function expansion method for the modi ed Korteweg-de Vries-Zakharov-Kuznetsov and the Hirota equations, Phys. Lett. A, 2001, 289, 69-74. Google Scholar

  • [13]

    Simbanefayi I., Khalique C.M., Travelling wave solutions and conservation laws for the Korteweg-de Vries-Bejamin-Bona-Mahony equation, Results in Physics, 2018, 8, 57-63 CrossrefWeb of ScienceGoogle Scholar

  • [14]

    Kudryashov N.A., One method for finding exact solutions of nonlinear differential equations, Comm. Nonlin. Sci. Numer. Simulat., 2012, 17, 2248-2253. CrossrefGoogle Scholar

  • [15]

    Motsepa T., Khalique C.M., Conservation laws and solutions of a generalized coupled (2+1)-dimensional Burgers system, Comp. Math. Appl., 2017, 74, 1333-1339. CrossrefGoogle Scholar

  • [16]

    Olver P.J., Applications of lie groups to differential equations, Springer-Verlag, New York, 1993. Google Scholar

  • [17]

    Ibragimov N.H. (Ed.), CRC Handbook of Lie group analysis of differential equations, 1994-1996, vol. 1-3, CRC Press, Boca Raton. Google Scholar

  • [18]

    Ibragimov N.H., Elementary Lie group analysis and ordinary differential equations, 1999, John Wiley and Sons, Chichester. Google Scholar

  • [19]

    Hydon P.E., Symmetry methods for differential equations, 2000, Cambridge, Cambridge University Press. Google Scholar

  • [20]

    Motsepa T., Khalique C.M., Gandarias M.L., Symmetry analysis and conservation laws of the Zoomeron equation, Symmetry, 2017, 9, 27. Web of ScienceCrossrefGoogle Scholar

  • [21]

    Motsepa T., Aziz T., Fatima A., Khalique C.M., Algebraic aspects of evolution partial differential equation arising in the study of constant elasticity of variance model from financial mathematics, Open Phys., 2018, 16, 31-36. CrossrefWeb of ScienceGoogle Scholar

  • [22]

    Naz R., Mahomed F.M., Mason D.P., Comparison of different approaches to conservation laws for some partial differential equations in fluid mechanics, Appl. Math. Comp., 2008, 205, 212-230. CrossrefGoogle Scholar

  • [23]

    Anco S.C., Bluman G., Direct construction method for conservation laws of partial differential equations. Part I: Examples of conservation law classifications, Eur. J. Pure Appl. Math., 2002, 13, 545-566. Google Scholar

  • [24]

    Anco S.C., Bluman G., Direct construction method for conservation laws of partial differential equations. Part II: General treatment, Eur. J. Pure Appl. Math., 2002, 13, 567-585. Google Scholar

  • [25]

    De la Rosa R., Bruzón M.S., On the classical and nonclassical symmetries of a generalized Gardner equation, Appl. Math. Nonlin. Sci., 2016, 1, 263-272. Google Scholar

  • [26]

    Rosa M., Gandarias M.L., Multiplier method and exact solutions for a density dependent reaction-diffusion equation, Appl. Math. Nonlin. Sci., 2016, 1, 311-320. Google Scholar

  • [27]

    Gandarias M.L., Bruzón M.S., Conservation laws for a Boussinesq equation, Appl. Math. Nonlin. Sci., 2017, 2, 465-472. Google Scholar

  • [28]

    Motsepa T., Khalique C.M., On the conservation laws and solutions of a (2+1) dimensional KdV-mKdV equation of mathematical physics, Open Phys., 2018, 16, 211-214. CrossrefWeb of ScienceGoogle Scholar

  • [29]

    Leveque R.J., Numerical methods for conservation laws, 1992, Birkhäuser, Basel. Google Scholar

  • [30]

    Burgers J.M., Application of a model system to illustrate some 20 points of statistical theory of free turbulence, Royal Netherlands Acad. Sci., XLIII 1940, 1, 2-12. Google Scholar

  • [31]

    Morrison P.J., Meiss J.D., Cary J.R., Scattering of regularized long- wave solitary waves, Physica, 1984, 11D, 324-336. Google Scholar

  • [32]

    Zaki S.I., A least-squares finite element scheme for the EW equa tion, Comp. Meth. Appl. Mech. Eng., 2000, 189, 587-594. CrossrefGoogle Scholar

  • [33]

    Zaki S.I., Solitary waves induced by the boundary forced EW equation, Comp. Meth. Appl. Mech. Eng., 2001, 190, 4881-4887. CrossrefGoogle Scholar

  • [34]

    Arora R., Siddiqui Md.J., Singh V.P., Solutions of invscid Burgers’ and equal width wave equations by RDTM, International J. Appl. Phys. Math., 2012, 2, 212-214.. Google Scholar

  • [35]

    Zaki S.I., Solitarywave interactions for the modified equalwidth equation, Comp. Phys. Comm., 2000, 126, 219-231. CrossrefGoogle Scholar

  • [36]

    Wazwaz A.M., The tanh and sine-cosine methods for a reliable treatment of the modified equalwidth equation and its variants, Comm.. Nonlin. Sci. Numer. Simul., 2006, 11, 148-160. CrossrefGoogle Scholar

  • [37]

    Hassan H.N., An accurate numerical solution for the modified equal width wave equation using the Fourier pseudo-spectral method, J. Appl. Math. Phys., 2016, 4, 1054-1067. CrossrefGoogle Scholar

  • [38]

    Evans D.J., Raslan K.R., Solitary waves for the generalized equal width (GEW) equation, Int. J. Comp. Math., 20005, 82, 445-455. Google Scholar

  • [39]

    Saha A., Bifurcation, periodic and chaotic motions of the modified equal width-Burgers (MEW-Burgers) equation with external periodic perturbation, Nonlin. Dyn., 2007, 87, 2193-2201. Google Scholar

  • [40]

    Ibragimov N.H., A new conservation theorem, J. Math. Anal. Appl., 2007, 333, 311-328. CrossrefGoogle Scholar

  • [41]

    Mothibi D.M., Khalique C.M., Conservation laws and exact solutions of a generalized Zakharov-Kuznetsov equation, Symmetry, 2015, 7, 949-961. CrossrefWeb of ScienceGoogle Scholar

About the article

Received: 2018-05-31

Accepted: 2018-07-02

Published Online: 2018-12-26

Citation Information: Open Physics, Volume 16, Issue 1, Pages 795–800, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2018-0099.

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©2018 Chaudry Masood Khalique and Innocent Simbanefayi, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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