August  2016, 21(6): 2057-2071. doi: 10.3934/dcdsb.2016036

Reduction and bifurcation of traveling waves of the KdV-Burgers-Kuramoto equation

1. 

School of Applied Mathematics, Chengdu University of Information Technology, Chengdu, Sichuan 610225, China

2. 

School of Computer Science and Technology, Southwest University for Nationalities, Chengdu, Sichuan 610041, China

Received  April 2015 Revised  April 2016 Published  June 2016

In this paper, the Lie symmetry analysis is performed on the KBK equation. By constructing its one-dimensional optimal system, we obtain four classes of reduced equations and corresponding group-invariant solutions. Particularly, the traveling wave equation, as an important reduced equation, is investigated in detail. Treating it as a singular perturbation system in $\mathbb{R}^3$, we study the phase space geometry of its reduced system on a two-dimensional invariant manifold by using the dynamical system methods such as tracking the unstable manifold of the saddle, studying the equilibria at infinity and discussing the homoclinic bifurcation and Poincaré bifurcation. Correspongding wavespeed conditions are determined to guarantee the existence of various bounded traveling waves of the KBK equation.
Citation: Yuqian Zhou, Qian Liu. Reduction and bifurcation of traveling waves of the KdV-Burgers-Kuramoto equation. Discrete and Continuous Dynamical Systems - B, 2016, 21 (6) : 2057-2071. doi: 10.3934/dcdsb.2016036
References:
[1]

G. W. Bluman and J. D. Cole, Similarity Methods for Differential Equation, Springer-Verlag, New York-Heidelberg, 1974. doi: 10.1007/978-1-4612-6394-4.

[2]

G. W. Bluman and S. Kumei, Symmetries and Differential Equations, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-4307-4.

[3]

S. N. Chow and J. K. Hale, Method of Bifurcation Theory, Springer-Verlag, New York, 1982.

[4]

E. G. Fan, Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A, 277 (2000), 212-218. doi: 10.1016/S0375-9601(00)00725-8.

[5]

N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations, 31 (1979), 53-98. doi: 10.1016/0022-0396(79)90152-9.

[6]

Z. T. Fu, S. K. Liu and S. D. Liu, New exact solutions to the KdV-Burgers-Kuramoto equation, Chaos. Soliton. Fract., 23 (2005), 609-616. doi: 10.1016/j.chaos.2004.05.012.

[7]

Y. G. Fu and Z. R. Liu, Persistence of travelling fronts of KdV-Burgers-Kuramoto equation, Appl. Math. comp., 216 (2010), 2199-2206. doi: 10.1016/j.amc.2010.03.057.

[8]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-1140-2.

[9]

J. G. Guo, L. J. Zhou and S. Y. Zhang, Geometrical nonlinear waves in finite deformation elastic rods, Appl. Math. Mech., 26 (2005), 667-674. doi: 10.1007/BF02466342.

[10]

A. K. Gupta and S. Saha Ray, Traveling wave solution of fractional KdV-Burger-Kuramoto equation describing nonlinear physical phenomena, AIP Advances, 4 (2014), 097120. doi: 10.1063/1.4895910.

[11]

N. H. Ibragimov, Transformation Groups Applied to Mathematical Physics, Reidel, Dordrecht, 1985. doi: 10.1007/978-94-009-5243-0.

[12]

C. K. R. T. Jones, Geometric singular perturbation, in Dynamical Systems, Springer Lecture Notes Math., 1609 (1995), 44-120. doi: 10.1007/BFb0095239.

[13]

B. Katzengruber, M. Krupa and P. Szmolyan, Bifurcation of traveling waves in extrinsic semiconductors, Physica D, 144 (2000), 1-19. doi: 10.1016/S0167-2789(00)00030-0.

[14]

T. Kawahara, Formation of saturated solitons in a nonlinear dispersive system with instability and dissipation, Phys. Rev. Lett., 51 (1983), 381-383. doi: 10.1103/PhysRevLett.51.381.

[15]

S. A. Khuri, Traveling wave solutions for nonlinear differential equations: A unified ansätze approach, Chaos. Soliton. Fract., 32 (2007), 252-258. doi: 10.1016/j.chaos.2005.10.106.

[16]

J. M. Kim and C. Chun, New exact solutions to the KdV-Burgers-Kuramoto equation with the Exp-function method, Abstr. Appl. Anal., 2012 (2012), Art. ID 892420, 10 pp. doi: 10.1155/2012/892420.

[17]

N. A. Kudryashov, Exact solutions of the generalized Kuramoto-Sivashinsky equation, Phys. Lett. A, 147 (1990), 287-291. doi: 10.1016/0375-9601(90)90449-X.

[18]

N. A. Kudryashov and E. D. Zargaryan, Solitary waves in active-dissipative dispersive media, J. Phys. A, 29 (1996), 8067-8077. doi: 10.1088/0305-4470/29/24/029.

[19]

K. L. Lan and H. B. Wang, Exact solutions for two nonlinear equations: I, J. Phys. A, 23 (1990), 3923-3928. doi: 10.1088/0305-4470/23/17/021.

[20]

J. B. Li and H. H. Dai, On the Study of Singular Nonlinear Travelling Wave Equation: Dynamical System Approach, Science Press, Beijing, 2007.

[21]

J. B. Li, Bifurcations and exact travelling wave solutions of the generalized two-component Hunter-Saxton system, Discrete Cont. Dyn.-B, 19 (2014), 1719-1729. doi: 10.3934/dcdsb.2014.19.1719.

[22]

J. B. Li and F. J. Chen, Exact traveling wave solutions and bifurcations of the dual Ito equation, Nonlinear Dynam., 82 (2015), 1537-1550. doi: 10.1007/s11071-015-2259-y.

[23]

C. Z. Li and Z. F. Zhang, A criterion for determining the monotonicity of the ratio of two Abelian integrals, J. Differential Equations, 124 (1996), 407-424. doi: 10.1006/jdeq.1996.0017.

[24]

H. Z. Liu, Comment on "New Exact Solutions to the KdV-Burgers-Kuramoto Equation with the Exp-Function Method", Abstr. Appl. Anal., 2014 (2014), Art. ID 240784, 4 pp. doi: 10.1155/2014/240784.

[25]

S. D. Liu, S. K. Liu, Z. H. Huang and Q. Zhao, On a class of nonlinear Schrödinger equations III, Prog. Natural Sci., 9 (1999), 912-918.

[26]

P. J. Olver, Application of Lie Groups to Differential Equations, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4684-0274-2.

[27]

L. V. Ovsiannikov, Group Analysis of Differential Equations, Academic, New York, 1982.

[28]

E. J. Parkes and B. R. Duffy, An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations, Comp. Phys. Commun., 98 (1996), 288-300. doi: 10.1016/0010-4655(96)00104-X.

[29]

S. A. Sezer, A. Yildirim and S. T. Mohyud-Din, He's homotopy perturbation method for solving the fractional KdV-Burgers-Kuramoto equation, Int. J. Numer. Method H., 21 (2011), 448-458. doi: 10.1108/09615531111123119.

[30]

G. I. Sivashinsky, Large cells in nonlinear marangoni convection, Physica D, 4 (1982), 227-235. doi: 10.1016/0167-2789(82)90063-X.

[31]

L. L. Wei, Y. N. He and A. Yildirim, Numerical algorithm based on an implicit fully discrete local discontinuous Galerkin method for the time-fractional KdV-Burgers-Kuramoto equation, Zamm-Z. Angew. Math. Me., 93 (2013), 14-28. doi: 10.1002/zamm.201200003.

[32]

Y. Xie, S. Zhu and K. Su, Solving the KdV-Burgers-Kuramoto equation by a combination method, Int. J. Modern Phys. B, 23 (2009), 2101-2106. doi: 10.1142/S0217979209052017.

[33]

E. Zeidler, Applied Functional Analysis, Springer-Verlag, New York, 1995.

[34]

S. Zhang, New exact solutions of the KdV-Burgers-Kuramoto equation, Phys. Lett. A, 358 (2006), 414-420. doi: 10.1016/j.physleta.2006.05.071.

[35]

Z. F. Zhang, T. R. Ding, W. Z. Huang and Z. X. Dong, Qualitative Theory of Differential Equations, Amer. Math. Soc., Providence, 1992.

show all references

References:
[1]

G. W. Bluman and J. D. Cole, Similarity Methods for Differential Equation, Springer-Verlag, New York-Heidelberg, 1974. doi: 10.1007/978-1-4612-6394-4.

[2]

G. W. Bluman and S. Kumei, Symmetries and Differential Equations, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-4307-4.

[3]

S. N. Chow and J. K. Hale, Method of Bifurcation Theory, Springer-Verlag, New York, 1982.

[4]

E. G. Fan, Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A, 277 (2000), 212-218. doi: 10.1016/S0375-9601(00)00725-8.

[5]

N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations, 31 (1979), 53-98. doi: 10.1016/0022-0396(79)90152-9.

[6]

Z. T. Fu, S. K. Liu and S. D. Liu, New exact solutions to the KdV-Burgers-Kuramoto equation, Chaos. Soliton. Fract., 23 (2005), 609-616. doi: 10.1016/j.chaos.2004.05.012.

[7]

Y. G. Fu and Z. R. Liu, Persistence of travelling fronts of KdV-Burgers-Kuramoto equation, Appl. Math. comp., 216 (2010), 2199-2206. doi: 10.1016/j.amc.2010.03.057.

[8]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-1140-2.

[9]

J. G. Guo, L. J. Zhou and S. Y. Zhang, Geometrical nonlinear waves in finite deformation elastic rods, Appl. Math. Mech., 26 (2005), 667-674. doi: 10.1007/BF02466342.

[10]

A. K. Gupta and S. Saha Ray, Traveling wave solution of fractional KdV-Burger-Kuramoto equation describing nonlinear physical phenomena, AIP Advances, 4 (2014), 097120. doi: 10.1063/1.4895910.

[11]

N. H. Ibragimov, Transformation Groups Applied to Mathematical Physics, Reidel, Dordrecht, 1985. doi: 10.1007/978-94-009-5243-0.

[12]

C. K. R. T. Jones, Geometric singular perturbation, in Dynamical Systems, Springer Lecture Notes Math., 1609 (1995), 44-120. doi: 10.1007/BFb0095239.

[13]

B. Katzengruber, M. Krupa and P. Szmolyan, Bifurcation of traveling waves in extrinsic semiconductors, Physica D, 144 (2000), 1-19. doi: 10.1016/S0167-2789(00)00030-0.

[14]

T. Kawahara, Formation of saturated solitons in a nonlinear dispersive system with instability and dissipation, Phys. Rev. Lett., 51 (1983), 381-383. doi: 10.1103/PhysRevLett.51.381.

[15]

S. A. Khuri, Traveling wave solutions for nonlinear differential equations: A unified ansätze approach, Chaos. Soliton. Fract., 32 (2007), 252-258. doi: 10.1016/j.chaos.2005.10.106.

[16]

J. M. Kim and C. Chun, New exact solutions to the KdV-Burgers-Kuramoto equation with the Exp-function method, Abstr. Appl. Anal., 2012 (2012), Art. ID 892420, 10 pp. doi: 10.1155/2012/892420.

[17]

N. A. Kudryashov, Exact solutions of the generalized Kuramoto-Sivashinsky equation, Phys. Lett. A, 147 (1990), 287-291. doi: 10.1016/0375-9601(90)90449-X.

[18]

N. A. Kudryashov and E. D. Zargaryan, Solitary waves in active-dissipative dispersive media, J. Phys. A, 29 (1996), 8067-8077. doi: 10.1088/0305-4470/29/24/029.

[19]

K. L. Lan and H. B. Wang, Exact solutions for two nonlinear equations: I, J. Phys. A, 23 (1990), 3923-3928. doi: 10.1088/0305-4470/23/17/021.

[20]

J. B. Li and H. H. Dai, On the Study of Singular Nonlinear Travelling Wave Equation: Dynamical System Approach, Science Press, Beijing, 2007.

[21]

J. B. Li, Bifurcations and exact travelling wave solutions of the generalized two-component Hunter-Saxton system, Discrete Cont. Dyn.-B, 19 (2014), 1719-1729. doi: 10.3934/dcdsb.2014.19.1719.

[22]

J. B. Li and F. J. Chen, Exact traveling wave solutions and bifurcations of the dual Ito equation, Nonlinear Dynam., 82 (2015), 1537-1550. doi: 10.1007/s11071-015-2259-y.

[23]

C. Z. Li and Z. F. Zhang, A criterion for determining the monotonicity of the ratio of two Abelian integrals, J. Differential Equations, 124 (1996), 407-424. doi: 10.1006/jdeq.1996.0017.

[24]

H. Z. Liu, Comment on "New Exact Solutions to the KdV-Burgers-Kuramoto Equation with the Exp-Function Method", Abstr. Appl. Anal., 2014 (2014), Art. ID 240784, 4 pp. doi: 10.1155/2014/240784.

[25]

S. D. Liu, S. K. Liu, Z. H. Huang and Q. Zhao, On a class of nonlinear Schrödinger equations III, Prog. Natural Sci., 9 (1999), 912-918.

[26]

P. J. Olver, Application of Lie Groups to Differential Equations, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4684-0274-2.

[27]

L. V. Ovsiannikov, Group Analysis of Differential Equations, Academic, New York, 1982.

[28]

E. J. Parkes and B. R. Duffy, An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations, Comp. Phys. Commun., 98 (1996), 288-300. doi: 10.1016/0010-4655(96)00104-X.

[29]

S. A. Sezer, A. Yildirim and S. T. Mohyud-Din, He's homotopy perturbation method for solving the fractional KdV-Burgers-Kuramoto equation, Int. J. Numer. Method H., 21 (2011), 448-458. doi: 10.1108/09615531111123119.

[30]

G. I. Sivashinsky, Large cells in nonlinear marangoni convection, Physica D, 4 (1982), 227-235. doi: 10.1016/0167-2789(82)90063-X.

[31]

L. L. Wei, Y. N. He and A. Yildirim, Numerical algorithm based on an implicit fully discrete local discontinuous Galerkin method for the time-fractional KdV-Burgers-Kuramoto equation, Zamm-Z. Angew. Math. Me., 93 (2013), 14-28. doi: 10.1002/zamm.201200003.

[32]

Y. Xie, S. Zhu and K. Su, Solving the KdV-Burgers-Kuramoto equation by a combination method, Int. J. Modern Phys. B, 23 (2009), 2101-2106. doi: 10.1142/S0217979209052017.

[33]

E. Zeidler, Applied Functional Analysis, Springer-Verlag, New York, 1995.

[34]

S. Zhang, New exact solutions of the KdV-Burgers-Kuramoto equation, Phys. Lett. A, 358 (2006), 414-420. doi: 10.1016/j.physleta.2006.05.071.

[35]

Z. F. Zhang, T. R. Ding, W. Z. Huang and Z. X. Dong, Qualitative Theory of Differential Equations, Amer. Math. Soc., Providence, 1992.

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