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Retracted: Bifurcation of travelling wave solutions of the generalized Zakharov equation. (English) Zbl 1436.35287

J. Appl. Math. 2014, Article ID 170946, 11 p. (2014); retraction ibid. 2016, Article ID 3176846, 1 p. (2016).
Summary: By using bifurcation theory of planar ordinary differential equations all different bounded travelling wave solutions of the generalized Zakharov equation are classified in to different parametric regions. In each of these parametric regions the exact explicit parametric representation of all solitary, kink (antikink), and periodic wave solutions as well as their numerical simulation and their corresponding phase portraits are obtained.
Editorial remark: This article has been retracted; see [ibid. 2016, Article ID 3176846, 1 p. (2016; Zbl 1436.35283)].

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)

Citations:

Zbl 1436.35283
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References:

[1] 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
[2] Hirota, R., The Direct Method in Soliton Theory. The Direct Method in Soliton Theory, Cambridge Tracts in Mathematics, 155 (2004), Cambridge, UK: Cambridge University Press, Cambridge, UK
[3] Ablowitz, M. J.; Clarkson, P. A., Soliton, Nonlinear Evolution Equations and Inverse Scattering. Soliton, Nonlinear Evolution Equations and Inverse Scattering, London Mathematical Society Lecture Note Series, 149 (1991), Cambridge, Mass, USA: Cambridge University Press, Cambridge, Mass, USA
[4] Rogers, C.; Schief, W. K., Bäcklund and Darboux Transformations, Geometry and Modern Applications in Soliton Theory. Bäcklund and Darboux Transformations, Geometry and Modern Applications in Soliton Theory, Cambridge Texts in Applied Mathematics (2002), Cambridge, UK: Cambridge University Press, Cambridge, UK
[5] Olver, P. J., Applications of Lie Groups to Differential Equations (1993), New York, NY, USA: Springer, New York, NY, USA
[6] Fan, E. G., Integrable Systems and Computer Algebra (2004), Science Press
[7] Yan, C., A simple transformation for nonlinear waves, Physics Letters A, 224, 1-2, 77-84 (1996) · Zbl 1037.35504
[8] He, J., Application of homotopy perturbation method to nonlinear wave equations, Chaos, Solitons and Fractals, 26, 3, 695-700 (2005) · Zbl 1072.35502
[9] Wang, M. L., Exact solutions for a compound KdV-Burgers equation, Physics Letters A: General, Atomic and Solid State Physics, 213, 5-6, 279-287 (1996) · Zbl 0972.35526
[10] Junqi, H., An algebraic method exactly solving two high-dimensional nonlinear evolution equations, Chaos, Solitons and Fractals, 23, 2, 391-398 (2005) · Zbl 1069.35065
[11] Liu, S.; Fu, Z.; Zhao, Q., Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Physics Letters A, 285, 1-2, 69-74 (2001) · Zbl 0972.35062
[12] Li, J.; Zhang, L., Bifurcations of traveling wave solutions in generalized Pochhammer-Chree equation, Chaos, Solitons and Fractals, 14, 4, 581-593 (2002) · Zbl 0997.35096
[13] Li, J.; He, T. L., Exact traveling wave solutions and bifurcations in a nonlinear elastic rod equation, Acta Mathematicae Applicatae Sinica, 26, 2, 283-306 (2010) · Zbl 1371.34002
[14] Zakharov, V. E.; Shabat, A. B., Exact theory of two-dimensional self-focusing and one dimensionsl self-modulation of waves in nonlinear media, Journal of Experimental and Theoretical Physics, 34, 1, 62-69 (1972)
[15] Kit, E.; Shemer, L., Spatial versions of the Zakharov and Dysthe evolution equations for deep-water gravity waves, Journal of Fluid Mechanics, 450, 201-205 (2002) · Zbl 1004.76011
[16] Goldman, M. V., Strong turbulence of plasma waves, Reviews of Modern Physics, 56, 4, 709-735 (1984)
[17] Anderson, D., Variational approach to nonlinear pulse propagation in optical fibers, Physical Review A, 27, 6, 3135-3145 (1983)
[18] Zakharov, V. E., Collapse of Langmuir waves, Soviet Physics JETP, 35, 5, 908-914 (1972)
[19] Zakharov, V. E.; Synakh, V. S., The nature of the self-focusing singularity, Soviet Physics-JETP, 41, 465-468 (1976)
[20] Hale, J. K.; Koçak, H., Dynamics and Bifurcation (1991), New York, NY, USA: Springer, New York, NY, USA
[21] Byrd, P. F.; Friedman, M. D., Handbook of Elliptic Integrals for Engineers and Scientists (1971), Berlin, Germany: Springer, Berlin, Germany
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