 Research
 Open Access
 Published:
Dynamic behavior of traveling wave solutions for new couplings of the Burgers equations with timedependent variable coefficients
Advances in Difference Equations volumeÂ 2017, ArticleÂ number:Â 167 (2017)
Abstract
In this paper, we develop the nonlinear integrable couplings of Burgers equations with timedependent variable coefficients. A new simplified bilinear method is used to obtain new multiplekink solutions and multiplesingularkink solutions for this system. The proposed system is a generalization model in ocean dynamics, plasma physics and nonlinear lattice. The effects of timevariable coefficients on the velocity, phase and amplitude are given. The solitonic propagation and collision are discussed by the graphical analysis and characteristicline method.
1 Introduction
The classical coupled Burgers equations (CBE) [1â€“3] with time t and space x derivatives are given by
where \(t>0\), x is a horizontal coordinate space and a, b are constants. The coupled Burgers equations (CBE) arise in a large number of applications in physics, engineering and mathematical problems. Some if these applications are plasma physics, fluid mechanics, optic, solid state physics, chemical physics, etc. [1â€“3]. Many researchers in applied mathematics give great attention to finding the analytical, approximation and exact solutions of CBE by different methods such as variational iteration method [4], AdomianPade technique [5], differential transformation method [2], exponential function method in rational form [6], homotopy analysis method [7], modified extended direct algebraic (MEDA) method [8], first integral method [9], reduced differential transform method [10] and the Hirota bilinear method [11].
In this paper, we develop the classical coupled Burgers equations (1) to derive nonlinear ncoupled Burgers equations with timevariable coefficients (ncBE) in the form
When \(n=2\), \(\alpha_{1}(t)=\alpha_{2}(t)=1\), \(\beta_{1}(t)=\beta _{2}(t)=2\), \(\gamma_{1}(t)=a\), and \(\gamma_{2}(t)=b\), the coupling (2) reduces to the classical coupled (1). The objectives of this work are the following:

1.
Derive a form of nonlinear ncoupled Burgers equations (2).

2.
Show that it has multiplekink solutions and multiplesingularkink solutions by using the Backlund transformations and simplified Hirotaâ€™s method [12â€“26].
In this study, we need the following conditions on (2):
where \(b_{j}\) are arbitrary constants.
Finally, we define the â€˜kinkâ€™ as a type of solitons which is in the form tanh, not tanh^{2}. In a kink, we take the limit when x approaches infinity. The answer is a constant, unlike solitons where the limit goes to 0. Solitons are solutions in the form of sech and \(\mathit{sech}^{2}\). The graph of the soliton is a wave which is positive. It is unlike the periodic solutions sine, cosine, etc. In trigonometric functions, waves go above and below the horizontal line [27].
This paper is organized as follows. A new Nkink solutions and Nsingularkink solutions for the ncBE system (2) are constructed in SectionsÂ 1 andÂ 2. The effect of the variable coefficients and the collision behavior and propagation properties are discussed in SectionÂ 3. Finally, conclusions are given in SectionÂ 4.
2 Multiplekink solutions for the ncBE system
In this section, we use the simplified bilinear method [28â€“30] to construct multiplekink solutions of ncBE system (2). If we substitute
into the linear terms of Eq. (2), we get the dispersion relation as follows:
Thus,
Assume that the multiplekink solutions of (2) are
For singlekink solutions, the \(a_{j}(X,T)\) is given by
Substitute Eqs (6) and (7) into Eq. (2), then solving for \(C_{1},C_{2},C_{3},\ldots,C_{n}\), the nonzero solution is given by
To obtain a numerical value of \(R_{j}\), we set the constraints \(\frac {\alpha_{j}(t)}{\beta_{j}(t)}=b_{j},j=1,2,3,\ldots,n\), where \(b_{j}\) are arbitrary constants. Now, substitute Eq. (8) into Eq. (6), to obtain the singlekink solutions for (2) as follows:
where
To obtain the twokink solutions, let
where \(\phi_{1j}(x,t)\) and \(\phi_{2j}(x,t)\) are defined in Eq. (5). Using Eqs (10) and (6) and substituting the results in Eq. (2), we obtain the value of phase shift by
Hence,
Substituting Eqs (11), (10) and (8) into Eq.Â (6), we obtain twokink solutions for Eq.Â (2)
The threesoliton solutions are determined by
where
Proceeding as before, we find
Then
Thus, the threekink solution for Eq. (2) is given by
To this point, we reach the fact that Eq. (2) is completely integrable and Nkink solutions exist for \(N\geq1\) [12, 15]. Moreover, we can obtain Nkink solutions as follows:
3 Multiplesingularkink solutions for the ncBE system
In order to obtain the singlesingularkink solutions of Eq.Â (2), we substitute
into the linear part of Eq. (2); as a result, we get
Assume that the singlesingularkink solutions of Eq. (2) are
where \(a_{j}(x,t)\) is given by
Substituting Eq. (14) into Eq. (2) and solving for \(C_{j}\), we get
Similarly, we set the constraints \(\frac{\alpha_{j}(t)}{\beta_{j}(t)}=b_{j},j=1,2,3,\ldots,n\), where \(b_{j}\) are arbitrary constants to obtain a numerical value of \(C_{j}\). Then the singlesingularkink solutions of Eq. (2) are
where
The twosingularkink solutions are obtained by setting
Substituting Eq. (16) into Eq. (13) and then in Eq. (2), we obtain the phase shift \(b_{12}\) as
Substitute Eqs (17), (16) and (15) into Eq. (13), then the twosingularkink solutions for Eq.Â (2) are
For threesingularkink solutions, we use
Proceeding as before, the threesingularkink solutions for Eq. (2) are given by
In general, we can set Nsingularkink solutions for Eq. (2) as
4 Stabilities and propagation characteristics of solitary waves
In this section, we discuss the effect of nonhomogeneities, namely, variable coefficients to the ncBE. The dispersion relation will be used to give the characteristic line and velocity v for every soliton. The soliton amplitude amp for \(w_{j}(x,t)\), \(j=1,2,3,\ldots,n\), can be expressed as
Using the characteristicline method [31, 32], the characteristic wedge for each solitary wave for \(w_{j}(x,t)\) is defined by
The velocity v of each solitary wave for \(w_{j}(x,t)\), \(j=1,2,3,\ldots,n\), is
The soliton amplitude amp depends on the variable coefficients \(\alpha _{j}(t)\) and \(\beta_{j}(t)\) but not on the variable coefficient \(\gamma _{j}(t)\), see FigureÂ 1. The propagation velocity of the solitary wave Eq. depends only on the coefficient functions \(\alpha_{j}(t)\). Moreover, we see that from (19), as the inequality \(s_{i}\alpha_{j}(t)>0\) holds, the soliton will move in the direction of positive xaxis.
In FigureÂ 2, we choose \(s_{1}=0.5\), \(s_{2}=0.75\), \(\alpha_{j}(t)=\frac{8t}{5\Gamma(1.8)}\) and \(\beta_{j}(t)=\frac{4t}{5\Gamma(1.8)}\). Then the characteristic curve of Eq. (18) is given by
Then the soliton reveals the parabolic type propagation trajectory with unalterable amplitude but continuously changeable velocity.
In FigureÂ 3, we choose \(s_{1}=0.5\), \(s_{2}=0.75\), \(\alpha_{j}(t)=\frac{7\sin t}{10\Gamma(1.4)}\) and \(\beta_{j}(t)=\frac{7\sin t}{20\Gamma(1.4)}\). Then the characteristic curve of Eq. (18) is given by
We see from FigureÂ 3 that the propagation trajectory of the soliton presents the periodicity oscillation.
In FiguresÂ 4 and 5, we use Eq. (12) to discuss the interaction between two solitonic waves in a nonhomogeneous situation. In FigureÂ 4 the interaction is called the overtaking coalescence. In this figure, we choose \(s_{1}=0.25,s_{2}=0.5\), \(\alpha_{j}(t)=\frac{\sqrt{\pi}}{2}(t^{2}+t1)\) and \(\beta_{j}(t)=\frac{\sqrt{\pi}}{2}(t^{2}t+1)\). The two fronts with the same propagation direction in xaxis coalesce into one large front in their interaction region of the \((x,t)\)plane, of which the amplitude amounts to two initial amplitudes. The front with faster velocity overtakes the slowvelocity one. In FigureÂ 5, we choose \(s_{1}=0.25\), \(s_{2}=0.5\), \(\alpha_{j}(t)=\frac {\sqrt{\pi}}{2}(t^{2}+t1)\) and \(\beta_{j}(t)=\frac{\sqrt{\pi}}{2}(t^{2}t+1)\). The interaction is called headon collision between one leftgoing soliton and one rightgoing soliton. Moreover, the directions of the solitary are controlled by the sign of velocity. It is clear that the amplitude and velocity after the collision of each soliton are not changed since the phase shift \(b_{12}=0\).
5 Conclusions
In this work, we obtain new Nkink solutions and Nsingularkink solutions for new couplings of the Burgers equations with timedependent variable coefficients (ncBE) by using the simplified Hirota method and Backlund transformations. The condition \(\alpha_{j}(t)=b_{j}\beta_{j}(t)\) for Eq. (2) is sufficient to have multisoliton solutions. We show the effect of timedependent coefficients on amplitude and velocity of a single wave. We see that the amplitude depends on \(\alpha_{j}(t)\) and \(\beta_{j}(t)\), but the velocity of the wave depends only on \(\alpha_{j}(t)\) and both of them are independent of \(\gamma_{j}(t)\). Furthermore, the interaction behaviors and propagation characteristics of the solitons have been discussed. We see that the forms of the variable coefficients determine the appearances of the characteristic curve and correspond to distinct propagation trajectories.
Since the problem of bidirectional solitary waves has been reported in waves, in bubbly liquids [33, 34] and shallowwater waves [32], it is expected that the bidirectional solitonlike solutions to Eq. (2) are used to describe such interesting physical phenomena.
Regarding the complexity of the proposed problem, we highlight the main advantages of the proposed method:

1.
The solution in the proposed method can be written in the exponential form, which generates multiple solutions, while other methods generate only single solution.

2.
The proposed method shows the integrability of the modified equations, which is not possible in other methods.

3.
In the proposed method, we use auxiliary functions to identify the type of the obtained solution, which is not possible in other methods.

4.
The computational cost for the proposed method is cheaper compared with other methods.
Finally, most of the solitary wave methods give only single solution, either of type soliton, singularsoliton, kink, singularkink, periodic or singularperiodic. Examples of these methods are the tanh expansion method, the sinecosine method, the rational trigonometric function method, the tanhsech function method, the \((G'/G)\)expansion method, Jacobi elliptic function method and others [35â€“41]. The obtained solutions are always single. But, for the bilinear method, it gives multiple solutions at once.
References
Nee, J, Duan, J: Limit set of trajectories of the coupled viscous Burgerâ€™s equations. Appl. Math. Lett. 11(1), 5761 (1998)
Abazari, R, Abazari, R: Numerical study of some coupled PDEs by using differential transformation method. Int. J. Math. Comput. Phys. Electr. Comput. Eng. 4(6), 641648 (2010)
Esipov, SE: Coupled Burgers equations: a model of polydispersive sedimentation. Phys. Rev. E 52, 37113718 (1995)
Abdoua, MA, Solimanb, AA: Variational iteration method for solving Burgerâ€™s and coupled Burgerâ€™s equations. J. Comput. Appl. Math. 181, 245251 (2005)
Dehghan, M, Hamidi, A, Shakourifar, M: The solution of coupled Burgerâ€™s equations using Adomianâ€“Pade technique. Appl. Math. Comput. 189(2), 10341047 (2007)
AbdulZahra, KA: Extended exponential function method in rational form for exact solution of coupled Burgers equation. J. Basrah Res. Sci. 38(1), 7278 (2012)
Alomari, AK, Noorani, MSM, Nazar, R: The homotopy analysis method for the exact solutions of the \(K(2, 2)\), Burgers and coupled Burgers equations. Appl. Math. Sci. 2(40), 19631977 (2008)
Soliman, AA: The modified extended direct algebraic method for solving nonlinear partial differential equations. Int. J. Nonlinear Sci. 6(2), 136144 (2008)
AlSaif, AJS, AbdulHussein, A: Generating exact solutions of twodimensional coupled Burgersâ€™ equations by the first integral method. Res. J. Phys. Appl. Sci. 1(2), 2933 (2012)
Kumar, A, Arora, R: Solutions of the coupled system of Burgersâ€™ equations and coupled KleinGordon equation by RDT method. Int. J. Adv. Appl. Math. Mech. 1(2), 133145 (2013)
Zuo, JM: The Hirota bilinear method for the coupled Burgers equation and the high order BoussinesqBurgers equation. Chin. Phys. B 20(1), 010205 (2011)
Hirota, R: Direct methods in soliton theory. In: Bullough, RK, Caudrey, PJ (eds.) Solitons. Springer, Berlin (1980)
Hirota, R: Exact Nsoliton solutions of a nonlinear wave equation. J. Math. Phys. 14(7), 805809 (1973)
Hirota, R: Exact solutions of the Kortewegde Vries equation for multiple collisions of solitons. Phys. Rev. Lett. 27, 11921194 (1971)
Hirota, R: Exact Nsoliton solutions of a nonlinear wave equation. J. Math. Phys. 14, 805809 (1973)
Hirota, R: Exact solution of the modified Kortewegde Vries equation for multiple collisions of solitons. J. Phys. Soc. Jpn. 33, 14561458 (1972)
Awawdeh, F, Jaradat, HM, AlSharaâ€™, S: Applications of a simplified bilinear method to ionacoustic solitary waves in plasma. Eur. Phys. J. D 66(40), 18 (2012)
Alquran, M, Jaradat, HM, AlSharaâ€™, S, Awawdeh, F: A new simplified bilinear method for the Nsoliton solutions for a generalized FmKdV equation with timedependent variable coefficients. Int. J. Nonlinear Sci. Numer. Simul. 16(6), 259269 (2015)
Jaradat, HM, Awawdeh, F, AlSharaâ€™, S, Alquran, M, Momani, S: Controllable dynamical behaviors and the analysis of fractal Burgers hierarchy with the full effects of inhomogeneities of media. Rom. J. Phys. 60, 324343 (2015)
Awawdeh, F, AlSharaâ€™, S, Jaradat, HM, Alomari, AK, Alshorman, R: Symbolic computation on soliton solutions for variable coefficient quantum ZakharovKuznetsov equation in magnetized dense plasmas. Int. J. Nonlinear Sci. Numer. Simul. 15(1), 3545 (2014)
Alsayyed, O, Jaradat, HM, Jaradat, MMM, Mustafa, Z, Shatat, F: Multisoliton solutions of the BBM equation arisen in shallow water. J. Nonlinear Sci. Appl. 9(4), 18071814 (2016)
Wazwaz, AM: Multiple soliton solutions for the \((2+1)\)dimensional asymmetric Nizhnik Novikov Veselov equation. Nonlinear Anal. 72, 13141318 (2010)
Wazwaz, AM: Completely integrable coupled KdV and coupled KP systems. Commun. Nonlinear Sci. Numer. Simul. 15, 28282835 (2010)
Wazwaz, AM: Multiplesoliton solutions for the Boussinesq equation. Appl. Math. Comput. 192, 479486 (2007)
Hereman, W, Zhuang, W: A macsyma program for the Hirota method, 13th World Congress. Comput. Appl. Math. 2, 842863 (1991)
Hietarinta, J: A search for bilinear equations passing Hirotaâ€™s threesoliton condition. II. mKdVtype bilinear equations. J. Math. Phys. 28, 20942101 (1987)
Alquran, M: Solitons and periodic solutions to nonlinear partial differential equations by the sinecosine method. Appl. Math. Inf. Sci. 6(1), 8588 (2012)
Jaradat, HM, AlSharaâ€™, S, Awawdeh, F, Alquran, M: Variable coefficient equations of the KadomtsevPetviashvili hierarchy: multiple soliton solutions and singular multiple soliton solutions. Phys. Scr. 85, 035001 (2012)
Jaradat, HM: New solitary wave and multiple soliton solutions for the timespace fractional Boussinesq equation. Ital. J. Pure Appl. Math. 36, 367376 (2016)
Jaradat, HM: Dynamic behavior of traveling wave solutions for a class for the timespace coupled fractional kdV system with timedependent coefficients. Ital. J. Pure Appl. Math. 36, 945958 (2016)
Veksler, A, Zarmi, Y: Wave interactions and the analysis of the perturbed Burgers equation. Physica D 211, 5773 (2005)
Yu, X, Gao, YT, Sun, ZY, Liu, Y: Nsoliton solutions, BÃ¤cklund transformation and Lax pair for a generalized variablecoefficient fifthorder Kortewegde Vries equation. Phys. Scr. 81, 045402 (2010)
Miksis, MJ, Tinq, L: Effective equations for multiphase flowswaves in bubbly liquid. Adv. Appl. Mech. 28, 141260 (1991)
Miksis, MJ, Tinq, L: Wave propagation in a bubbly liquid at small volume fraction. Chem. Eng. Commun. 118, 5973 (1992)
Krishnan, EV: Remarks on a system of coupled nonlinear wave equations. J. Math. Phys. 31, 11551156 (1990)
Alquran, M, Qawasmeh, A: Soliton solutions of shallow water wave equations by means of \((G^{\prime}/G)\)expansion method. J. Appl. Anal. Comput. 4(3), 221229 (2014)
Qawasmeh, A, Alquran, M: Reliable study of some new fifthorder nonlinear equations by means of \((G^{\prime}/G)\)expansion method and rational sinecosine method. Appl. Math. Sci. 8(120), 59855994 (2014)
Shukri, S, AlKhaled, K: The extended tanh method for solving systems of nonlinear wave equations. Appl. Math. Comput. 217(5), 19972006 (2010)
Qawasmeh, A, Alquran, M: Soliton and periodic solutions for \((2+1)\)dimensional dispersive long waterwave system. Appl. Math. Sci. 8(50), 24552463 (2014)
Alquran, M, Ali, M, AlKhaled, K: Solitary wave solutions to shallow water waves arising in fluid dynamics. Nonlinear Stud. 19(4), 555562 (2012)
Alquran, M, Qawasmeh, A: Classifications of solutions to some generalized nonlinear evolution equations and systems by the sinecosine method. Nonlinear Stud. 20(2), 263272 (2013)
Acknowledgements
The author would like to express his sincere gratitude to the editor and the reviewers for their valuable comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The author declares that they have no competing interests.
Publisherâ€™s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Jaradat, H.M. Dynamic behavior of traveling wave solutions for new couplings of the Burgers equations with timedependent variable coefficients. Adv Differ Equ 2017, 167 (2017). https://doi.org/10.1186/s1366201712231
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s1366201712231
Keywords
 Hirota bilinear method
 multiplekink solutions
 coupled Burgers equations