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
Hirota bilinear method multiple-kink solutions coupled Burgers equations1 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
References
- Nee, J, Duan, J: Limit set of trajectories of the coupled viscous Burger’s equations. Appl. Math. Lett. 11(1), 57-61 (1998) MathSciNetView ArticleMATHGoogle Scholar
- 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), 641-648 (2010) MATHGoogle Scholar
- Esipov, SE: Coupled Burgers equations: a model of polydispersive sedimentation. Phys. Rev. E 52, 3711-3718 (1995) View ArticleGoogle Scholar
- Abdoua, MA, Solimanb, AA: Variational iteration method for solving Burger’s and coupled Burger’s equations. J. Comput. Appl. Math. 181, 245-251 (2005) MathSciNetView ArticleGoogle Scholar
- Dehghan, M, Hamidi, A, Shakourifar, M: The solution of coupled Burger’s equations using Adomian–Pade technique. Appl. Math. Comput. 189(2), 1034-1047 (2007) MathSciNetMATHGoogle Scholar
- Abdul-Zahra, KA: Extended exponential function method in rational form for exact solution of coupled Burgers equation. J. Basrah Res. Sci. 38(1), 72-78 (2012) Google Scholar
- 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), 1963-1977 (2008) MathSciNetMATHGoogle Scholar
- Soliman, AA: The modified extended direct algebraic method for solving nonlinear partial differential equations. Int. J. Nonlinear Sci. 6(2), 136-144 (2008) MathSciNetMATHGoogle Scholar
- Al-Saif, AJS, Abdul-Hussein, A: Generating exact solutions of two-dimensional coupled Burgers’ equations by the first integral method. Res. J. Phys. Appl. Sci. 1(2), 29-33 (2012) Google Scholar
- Kumar, A, Arora, R: Solutions of the coupled system of Burgers’ equations and coupled Klein-Gordon equation by RDT method. Int. J. Adv. Appl. Math. Mech. 1(2), 133-145 (2013) MATHGoogle Scholar
- Zuo, J-M: The Hirota bilinear method for the coupled Burgers equation and the high order Boussinesq-Burgers equation. Chin. Phys. B 20(1), 010205 (2011) View ArticleGoogle Scholar
- Hirota, R: Direct methods in soliton theory. In: Bullough, RK, Caudrey, PJ (eds.) Solitons. Springer, Berlin (1980) Google Scholar
- Hirota, R: Exact N-soliton solutions of a nonlinear wave equation. J. Math. Phys. 14(7), 805-809 (1973) MathSciNetView ArticleMATHGoogle Scholar
- Hirota, R: Exact solutions of the Korteweg-de Vries equation for multiple collisions of solitons. Phys. Rev. Lett. 27, 1192-1194 (1971) View ArticleMATHGoogle Scholar
- Hirota, R: Exact N-soliton solutions of a nonlinear wave equation. J. Math. Phys. 14, 805-809 (1973) MathSciNetView ArticleMATHGoogle Scholar
- Hirota, R: Exact solution of the modified Korteweg-de Vries equation for multiple collisions of solitons. J. Phys. Soc. Jpn. 33, 1456-1458 (1972) View ArticleGoogle Scholar
- Awawdeh, F, Jaradat, HM, Al-Shara’, S: Applications of a simplified bilinear method to ion-acoustic solitary waves in plasma. Eur. Phys. J. D 66(40), 1-8 (2012) Google Scholar
- Alquran, M, Jaradat, HM, Al-Shara’, S, Awawdeh, F: A new simplified bilinear method for the N-soliton solutions for a generalized FmKdV equation with time-dependent variable coefficients. Int. J. Nonlinear Sci. Numer. Simul. 16(6), 259-269 (2015) MathSciNetGoogle Scholar
- Jaradat, HM, Awawdeh, F, Al-Shara’, 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, 324-343 (2015) Google Scholar
- Awawdeh, F, Al-Shara’, S, Jaradat, HM, Alomari, AK, Alshorman, R: Symbolic computation on soliton solutions for variable coefficient quantum Zakharov-Kuznetsov equation in magnetized dense plasmas. Int. J. Nonlinear Sci. Numer. Simul. 15(1), 35-45 (2014) MathSciNetView ArticleGoogle Scholar
- Alsayyed, O, Jaradat, HM, Jaradat, MMM, Mustafa, Z, Shatat, F: Multi-soliton solutions of the BBM equation arisen in shallow water. J. Nonlinear Sci. Appl. 9(4), 1807-1814 (2016) MathSciNetMATHGoogle Scholar
- Wazwaz, AM: Multiple soliton solutions for the \((2+1)\)-dimensional asymmetric Nizhnik Novikov Veselov equation. Nonlinear Anal. 72, 1314-1318 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Wazwaz, AM: Completely integrable coupled KdV and coupled KP systems. Commun. Nonlinear Sci. Numer. Simul. 15, 2828-2835 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Wazwaz, AM: Multiple-soliton solutions for the Boussinesq equation. Appl. Math. Comput. 192, 479-486 (2007) MathSciNetMATHGoogle Scholar
- Hereman, W, Zhuang, W: A macsyma program for the Hirota method, 13th World Congress. Comput. Appl. Math. 2, 842-863 (1991) Google Scholar
- Hietarinta, J: A search for bilinear equations passing Hirota’s three-soliton condition. II. mKdV-type bilinear equations. J. Math. Phys. 28, 2094-2101 (1987) MathSciNetView ArticleMATHGoogle Scholar
- Alquran, M: Solitons and periodic solutions to nonlinear partial differential equations by the sine-cosine method. Appl. Math. Inf. Sci. 6(1), 85-88 (2012) MathSciNetMATHGoogle Scholar
- Jaradat, HM, Al-Shara’, S, Awawdeh, F, Alquran, M: Variable coefficient equations of the Kadomtsev-Petviashvili hierarchy: multiple soliton solutions and singular multiple soliton solutions. Phys. Scr. 85, 035001 (2012) View ArticleMATHGoogle Scholar
- Jaradat, HM: New solitary wave and multiple soliton solutions for the time-space fractional Boussinesq equation. Ital. J. Pure Appl. Math. 36, 367-376 (2016) MathSciNetMATHGoogle Scholar
- Jaradat, HM: Dynamic behavior of traveling wave solutions for a class for the time-space coupled fractional kdV system with time-dependent coefficients. Ital. J. Pure Appl. Math. 36, 945-958 (2016) MathSciNetMATHGoogle Scholar
- Veksler, A, Zarmi, Y: Wave interactions and the analysis of the perturbed Burgers equation. Physica D 211, 57-73 (2005) MathSciNetView ArticleMATHGoogle Scholar
- Yu, X, Gao, YT, Sun, ZY, Liu, Y: N-soliton solutions, Bäcklund transformation and Lax pair for a generalized variable-coefficient fifth-order Korteweg-de Vries equation. Phys. Scr. 81, 045402 (2010) View ArticleMATHGoogle Scholar
- Miksis, MJ, Tinq, L: Effective equations for multiphase flows-waves in bubbly liquid. Adv. Appl. Mech. 28, 141-260 (1991) MathSciNetView ArticleGoogle Scholar
- Miksis, MJ, Tinq, L: Wave propagation in a bubbly liquid at small volume fraction. Chem. Eng. Commun. 118, 59-73 (1992) View ArticleGoogle Scholar
- Krishnan, EV: Remarks on a system of coupled nonlinear wave equations. J. Math. Phys. 31, 1155-1156 (1990) MathSciNetView ArticleMATHGoogle Scholar
- 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), 221-229 (2014) MathSciNetMATHGoogle Scholar
- Qawasmeh, A, Alquran, M: Reliable study of some new fifth-order nonlinear equations by means of \((G^{\prime}/G)\)-expansion method and rational sine-cosine method. Appl. Math. Sci. 8(120), 5985-5994 (2014) Google Scholar
- Shukri, S, Al-Khaled, K: The extended tanh method for solving systems of nonlinear wave equations. Appl. Math. Comput. 217(5), 1997-2006 (2010) MathSciNetMATHGoogle Scholar
- Qawasmeh, A, Alquran, M: Soliton and periodic solutions for \((2+1)\)-dimensional dispersive long water-wave system. Appl. Math. Sci. 8(50), 2455-2463 (2014) MathSciNetGoogle Scholar
- Alquran, M, Ali, M, Al-Khaled, K: Solitary wave solutions to shallow water waves arising in fluid dynamics. Nonlinear Stud. 19(4), 555-562 (2012) MathSciNetMATHGoogle Scholar
- Alquran, M, Qawasmeh, A: Classifications of solutions to some generalized nonlinear evolution equations and systems by the sine-cosine method. Nonlinear Stud. 20(2), 263-272 (2013) MathSciNetMATHGoogle Scholar