The bifurcation and exact peakons, solitary and periodic wave solutions for the Kudryashov-Sinelshchikov equation. (English) Zbl 1248.35174
Summary: In this paper, the Kudryashov-Sinelshchikov equation is studied by using the bifurcation method of dynamical systems and the method of phase portraits analysis. From a dynamic point of view, the existence of peakon, solitary wave, smooth and non-smooth periodic waves is proved and the sufficient conditions to guarantee the existence of the above solutions in different regions of the parametric space are given. Also, some new exact travelling wave solutions are presented through some special phase orbits.
MSC:
| 35Q51 | Soliton equations |
| 35C08 | Soliton solutions |
| 35C07 | Traveling wave solutions |
| 35B32 | Bifurcations in context of PDEs |
Keywords:
Kudryashov-Sinelshchikov equation; bifurcation; soliton; periodic wave; exact travelling wave solutionReferences:
| [1] | Kudryashov, N. A.; Sinelshchikov, D. I., Nonlinear waves in bubbly liquids with consideration for viscosity and heat transfer, Phys Lett A, 374, 2011-2016 (2010) · Zbl 1236.76075 |
| [2] | Kudryashov, N. A.; Sinelshchikov, D. I., Nonlinear evolution equations for describing waves in bubbly liquids with viscosity and heat transfer consideration, Appl Math Comput, 217, 414-421 (2010) · Zbl 1273.76407 |
| [3] | Kudryashov, N. A.; Sinelshchikov, D. I., Nonlinear waves in liquids with gas bubbles with account of viscosity and heat transfer, Fluid Dynam, 45, 96-112 (2010) · Zbl 1215.76103 |
| [4] | Randrüüt, M., On the Kudryashov-Sinelshchikov equation for waves in bubbly liquids, Phys Lett A, 375, 3687-3692 (2011) · Zbl 1252.76016 |
| [5] | Ryabov, P. N., Exact solutions of the Kudryashov and Sinelshchikov equation, Appl Math Comput, 217, 3585-3590 (2010) · Zbl 1205.35272 |
| [6] | Kudryashov, N. A., One method for finding exact solutions of nonlinear differential equations, Commun Nonlinear Sci Numer Simulat, 17, 2248-2253 (2012) · Zbl 1250.35055 |
| [7] | Ryabov, P. N.; Sinelshchikov, D. I.; Kochanov, M. B., Application of the Kudryashov method for finding exact solutions of the high order nonlinear evolution equations, Appl Math Comput, 218, 3965-3972 (2011) · Zbl 1246.35015 |
| [8] | Li, J.; Dai, H., On the study of singular nonlinear traveling wave equation: dynamical system approach (2007), Science Press: Science Press Bejing |
| [9] | Li, J.; Zhang, Y.; Chen, G., Exact solutions and their dynamics of traveling waves in three typical nonlinear wave equations, Int J Bifur Chaos, 7, 2249-2266 (2009) · Zbl 1176.34004 |
| [10] | He, B.; Li, J.; Long, Y.; Rui, W., Bifurcations of travelling wave solutions for a variant of Camassa-Holm equation, Nonlinear Anal RWA, 9, 222-232 (2008) · Zbl 1185.35217 |
| [11] | Li, J.; Liu, Z., Smooth and non-smooth traveling waves in a nonlinearly dispersive equation, Appl Math Model, 25, 41-56 (2000) · Zbl 0985.37072 |
| [12] | Byrd, P. F.; Friedman, M. D., Handbook of elliptic integrals for engineers and physicists (1971), Springer Verlag · Zbl 0213.16602 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
