Dynamic behavior of traveling wave solutions for new couplings of the Burgers equations with time-dependent variable coefficients
- Husein M Jaradat^{1, 2}Email author
https://doi.org/10.1186/s13662-017-1223-1
© The Author(s) 2017
Received: 29 December 2016
Accepted: 30 May 2017
Published: 14 June 2017
Abstract
In this paper, we develop the nonlinear integrable couplings of Burgers equations with time-dependent variable coefficients. A new simplified bilinear method is used to obtain new multiple-kink solutions and multiple-singular-kink solutions for this system. The proposed system is a generalization model in ocean dynamics, plasma physics and nonlinear lattice. The effects of time-variable coefficients on the velocity, phase and amplitude are given. The solitonic propagation and collision are discussed by the graphical analysis and characteristic-line method.
Keywords
1 Introduction
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 N-kink solutions and N-singular-kink solutions for the nc-BE 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.
2 Multiple-kink solutions for the nc-BE system
3 Multiple-singular-kink solutions for the nc-BE system
4 Stabilities and propagation characteristics of solitary waves
5 Conclusions
In this work, we obtain new N-kink solutions and N-singular-kink solutions for new couplings of the Burgers equations with time-dependent variable coefficients (nc-BE) 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 multi-soliton solutions. We show the effect of time-dependent 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 shallow-water waves [32], it is expected that the bidirectional soliton-like solutions to Eq. (2) are used to describe such interesting physical phenomena.
- 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, singular-soliton, kink, singular-kink, periodic or singular-periodic. Examples of these methods are the tanh expansion method, the sine-cosine method, the rational trigonometric function method, the tanh-sech 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.
Declarations
Acknowledgements
The author would like to express his sincere gratitude to the editor and the reviewers for their valuable comments.
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.
Authors’ Affiliations
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