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Dynamic behavior of traveling wave solutions for new couplings of the Burgers equations with timedependent variable coefficients
Advances in Difference Equations volume 2017, Article number: 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.
Introduction
The classical coupled Burgers equations (CBE) [1–3] 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. [1–3]. 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 Hirota’s method [12–26].
In this study, we need the following conditions on (2):
where \(b_{j}\) are arbitrary constants.
Finally, we define the ‘kink’ 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 Sections 1 and 2. The effect of the variable coefficients and the collision behavior and propagation properties are discussed in Section 3. Finally, conclusions are given in Section 4.
Multiplekink solutions for the ncBE system
In this section, we use the simplified bilinear method [28–30] 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. (6), we obtain twokink solutions for Eq. (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:
Multiplesingularkink solutions for the ncBE system
In order to obtain the singlesingularkink solutions of Eq. (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. (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
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 Figure 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 Figure 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 Figure 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 Figure 3 that the propagation trajectory of the soliton presents the periodicity oscillation.
In Figures 4 and 5, we use Eq. (12) to discuss the interaction between two solitonic waves in a nonhomogeneous situation. In Figure 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 Figure 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 \(b_{12}=0\).
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 [35–41]. The obtained solutions are always single. But, for the bilinear method, it gives multiple solutions at once.
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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
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Keywords
 Hirota bilinear method
 multiplekink solutions
 coupled Burgers equations