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Seadawy, A. R.; El-Rashidy, K.

Classification of multiply travelling wave solutions for coupled Burgers, combined KdV-modified KdV, and Schrödinger-KdV equations. (English) Zbl 1356.35208

Abstr. Appl. Anal. 2015, Article ID 369294, 7 p. (2015).
Summary: Some explicit travelling wave solutions to constructing exact solutions of nonlinear partial differential equations of mathematical physics are presented. By applying a theory of Frobenius decompositions and, more precisely, by using a transformation method to the coupled Burgers, combined Korteweg-de Vries- (KdV-) modified KdV and Schrödinger-KdV equation is written as bilinear ordinary differential equations and two solutions to describing nonlinear interaction of travelling waves are generated. The properties of the multiple travelling wave solutions are shown by some figures. All solutions are stable and have applications in physics.

Cited in 3 Documents

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35C07 Traveling wave solutions
35Q55 NLS equations (nonlinear Schrödinger equations)
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
35B35 Stability in context of PDEs

Keywords:

coupled Burgers; combined Korteweg-de Vries- (KdV-) modified KdV; Schrödinger-KdV; bilinear ordinary differential equations; travelling waves
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\textit{A. R. Seadawy} and \textit{K. El-Rashidy}, Abstr. Appl. Anal. 2015, Article ID 369294, 7 p. (2015; Zbl 1356.35208)
Full Text: DOI

References:

[1] Burgers, J. M., A mathematical model illustrating the theory of turbulence, Advances in Applied Mechanics, 171-199 (1948), New York, NY, USA: Academic Press, New York, NY, USA
[2] Fletcher, C. A. J., Generating exact solutions of the two-dimensional Burgers’ equations, International Journal for Numerical Methods in Fluids, 3, 3, 213-216 (1983) · Zbl 0563.76082
[3] Cole, J. D., On a quasi-linear parabolic equation occurring in aerodynamics, Quarterly of Applied Mathematics, 9, 225-236 (1951) · Zbl 0043.09902
[4] Olver, P. J., Applications of Lie Groups to Differential Equations, 107 (1986), New York, NY, USA: Springer, New York, NY, USA · Zbl 0588.22001
[5] Bluman, G.; Kumei, S., Symmetries and Differenti al Equations (1986), New York, NY, USA: Springer, New York, NY, USA
[6] Zhang, J.; Wu, F.; Shi, J., Simple soliton solution method for the combined KdV and MKdV equation, International Journal of Theoretical Physics, 39, 6, 1697-1702 (2000) · Zbl 0958.35124
[7] Seadawy, A. R., New exact solutions for the KdV equation with higher order nonlinearity by using the variational method, Computers & Mathematics with Applications, 62, 10, 3741-3755 (2011) · Zbl 1236.35156
[8] Seadawy, A. R., Stability analysis for Zakharov-Kuznetsov equation of weakly nonlinear ion-acoustic waves in a plasma, Computers & Mathematics with Applications, 67, 1, 172-180 (2014) · Zbl 1381.82023
[9] Agrawal, G. P., Applications of Nonlinear Fiber Optics (2008), Elsevier
[10] Infeld, E.; Rowlands, G., Nonlinear Waves, Solitons and Chaos (2000), Cambridge University Press · Zbl 0994.76001
[11] Khater, A. H.; Helal, M. A.; Seadawy, A. R., General soliton solutions of ndimensional nonlinear Schrödinger equation, IL Nuovo Cimento B, 115, 1303-1312 (2000)
[12] Seadawy, A. R., Exact solutions of a two-dimensional nonlinear Schrödinger equation, Applied Mathematics Letters, 25, 4, 687-691 (2012) · Zbl 1241.35191
[13] Liu, S.; Fu, Z.; Liu, S.; Zhao, Q., Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Physics Letters. A, 289, 1-2, 69-74 (2001) · Zbl 0972.35062
[14] Yan, Z.; Zhang, H., New explicit solitary wave solutions and periodic wave solutions for Whitham-Broer-Kaup equation in shallow water, Physics Letters A, 285, 5-6, 355-362 (2001) · Zbl 0969.76518
[15] Rogers, C.; Shadwick, W. F., Backlund Transformations and Their Applications (1982), New York, NY, USA: Academic Press, New York, NY, USA · Zbl 0492.58002
[16] Hirota, R., Exact solution of the korteweg-de vries equation for multiple Collisions of solitons, Physical Review Letters, 27, 18, 1192-1194 (1971) · Zbl 1168.35423
[17] He, J. H., Variational iteration method for delay differential equations, Communications in Nonlinear Science and Numerical Simulation, 2, 4, 235-236 (1997)
[18] Arife, A. S.; Yildirim, A., New modified variational iteration transform method (MVITM) for solving eighth-order boundary value problems in one step, World Applied Sciences Journal, 13, 10, 2186-2190 (2011)
[19] Helal, M. A.; Seadawy, A. R., Variational method for the derivative nonlinear Schrödinger equation with computational applications, Physica Scripta, 80, 3, 350-360 (2009) · Zbl 1179.35311
[20] Mohyud-Din, S. T., Modified variational iteration method for integro-differential equations and coupled systems, Zeitschrift für Naturforschung A, 65, 4, 277-284 (2010)
[21] Malfliet, W., Solitary wave solutions of nonlinear wave equations, The American Journal of Physics, 60, 7, 650-654 (1992) · Zbl 1219.35246
[22] Bekir, A.; Cevikel, A. C., Solitary wave solutions of two nonlinear physical models by tanh-coth method, Communications in Nonlinear Science and Numerical Simulation, 14, 5, 1804-1809 (2009)
[23] Seadawy, A. R.; El-Rashidy, K., Traveling wave solutions for some coupled nonlinear evolution equations, Mathematical and Computer Modelling, 57, 5-6, 1371-1379 (2013)
[24] Seadawy, A. R., Traveling wave solutions of the Boussinesq and generalized fifth order KdV equations by using the direct algebraic method, Applied Mathematical Sciences, 6, 4081-4090 (2012) · Zbl 1264.35083
[25] Salas, A. H.; Gomez, C. A., Application of the Cole-Hopf transformation for finding exact solutions to several forms of the seventh-order KdV equation, Mathematical Problems in Engineering, 2010 (2010) · Zbl 1191.35236
[26] Ma, W. X.; Fuchssteiner, B., Explicit and exact solutions to a Kolmogorov-Petrovskii-PISkunov equation, International Journal of Non-Linear Mechanics, 31, 3, 329-338 (1996) · Zbl 0863.35106
[27] Naher, H.; Abdullah, F. A.; Akbar, M. A., The exp-function method for new exact solutions of the nonlinear partial differential equations, International Journal of Physical Sciences, 6, 29, 6706-6716 (2011)
[28] Aslan, I.; Marinakis, V., Some remarks on exp-function method and its applications, Communications in Theoretical Physics, 56, 3, 397-403 (2011) · Zbl 1247.35013
[29] Yıldırım, A.; Pınar, Z., Application of the exp-function method for solving nonlinear reaction-diffusion equations arising in mathematical biology, Computers & Mathematics with Applications, 60, 7, 1873-1880 (2010) · Zbl 1205.35325
[30] Wazwaz, A.-M., A new (2+1)-dimensional Korteweg-de Vries equation and its extension to a new (3+1)-dimensional Kadomtsev-Petviashvili equation, Physica Scripta, 84, 3 (2011) · Zbl 1260.35191
[31] Wang, M.; Li, X.; Zhang, J., The \(\left(G' / G\right)\)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics, Physics Letters A, 372, 4, 417-423 (2008) · Zbl 1217.76023
[32] Kaya, D., An explicit solution of coupled viscous Burgers’ equation by the decomposition method, International Journal of Mathematics and Mathematical Sciences, 27, 11, 675-680 (2001) · Zbl 0997.35077
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