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December  2016, 9(6): 1629-1645. doi: 10.3934/dcdss.2016067

The bifurcations of solitary and kink waves described by the Gardner equation

Yiren Chen 1, and  Zhengrong Liu 2,

1. 

School of Mathematics, South China University of Technology, Guangzhou 510640, China
2. 

Department of Mathematics, South China University of Technology, Guangzhou 510640

Received  July 2015 Revised  September 2016 Published  November 2016

In this paper, we investigate the bifurcations of nonlinear waves described by the Gardner equation $u_{t}+a u u_{x}+b u^{2} u_{x}+\gamma u_{xxx}=0$. We obtain some new results as follows: For arbitrary given parameters $b$ and $\gamma$, we choose the parameter $a$ as bifurcation parameter. Through the phase analysis and explicit expressions of some nonlinear waves, we reveal two kinds of important bifurcation phenomena. The first phenomenon is that the solitary waves with fractional expressions can be bifurcated from three types of nonlinear waves which are solitary waves with hyperbolic expression and two types of periodic waves with elliptic expression and trigonometric expression respectively. The second phenomenon is that the kink waves can be bifurcated from the solitary waves and the singular waves.
Keywords: kink waves., solitary waves, The Gardner equation, bifurcations, phase analysis.
Mathematics Subject Classification: Primary: 34A20, 34C35, 35B65; Secondary: 58F05, 76B2.
Citation: Yiren Chen, Zhengrong Liu. The bifurcations of solitary and kink waves described by the Gardner equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1629-1645. doi: 10.3934/dcdss.2016067
References:
[1]

G. Betchewe, K. K. Victor, B. B. Thomas and K. T. Crepin, New solutions of the Gardner equation: Analytical and numerical analysis of its dynamical understanding, Appl. Math. Comput., 223 (2013), 377-388. doi: 10.1016/j.amc.2013.08.028.  Google Scholar

[2]

A. Biswas and M. Song, Soliton solution and bifurcation analysis of the Zakharov-Kuznetsov-Benjamin-Bona-Mahoney equation with power law nonlinearity, Commun. Nonlinear Sci. Numer. Simulat., 18 (2013), 1676-1683. doi: 10.1016/j.cnsns.2012.11.014.  Google Scholar

[3]

Y. R. Chen and R. Liu, Some new nonlinear wave solutions for two (3+1)-dimensional equations, Appl. Math. Comput., 260 (2015), 397-411. doi: 10.1016/j.amc.2015.03.098.  Google Scholar

[4]

M. W. Coffey, On series expansions giving closed-form solutions of Korteweg-de Vries-like equations, SIAM J. Appl. Math., 50 (1990), 1580-1592. doi: 10.1137/0150093.  Google Scholar

[5]

Z. Fu, S. Liu and S. Liu, New kinds of solutions to Gardner equation, Chaos Soliton. Fract., 20 (2004), 301-309. doi: 10.1016/S0960-0779(03)00383-7.  Google Scholar

[6]

R. Grimshaw, D. Pelinovsky, E. Pelinovsky and A. Slunyaev, Generation of large-amplitude solitons in the extended Korteweg-de Vries equation, Chaos, 12 (2002), 1070-1076. doi: 10.1063/1.1521391.  Google Scholar

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R. Grimshawa, D. Pelinovsky, E. Pelinovsky and T. Talipova, Wave group dynamics in weakly nonlinear long-wave models, Physica D, 159 (2001), 35-57. doi: 10.1016/S0167-2789(01)00333-5.  Google Scholar

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R. Grimshaw, A. Slunyaev and E. Pelinovsky, Generation of solitons and breathers in the extended Korteweg-de Vries equation with positive cubic nonlinearity, Chaos, 20 (2010), 013102, 11pp. doi: 10.1063/1.3279480.  Google Scholar

[9]

K. Konno and Y. H. Ichikawa, A modified Korteweg de Vries equation for ion acoustic waves, J. Phys. Soc. Jpn., 37 (1974), 1631-1636. doi: 10.1143/JPSJ.37.1631.  Google Scholar

[10]

S. Liu, Z. Fu, S. Liu and Q. Zhao, Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Phys. Lett. A, 289 (2001), 69-74. doi: 10.1016/S0375-9601(01)00580-1.  Google Scholar

[11]

R. Liu and W. F. Yan, Some common expressions and new bifurcation phenomena for nonlinear waves in a generalized mKdV equation, Int. J. Bifurcat. Chaos, 23 (2013), 1330007, 19pp. doi: 10.1142/S0218127413300073.  Google Scholar

[12]

Y. Long, W. G. Rui and B. He, Travelling wave solutions for a higher order wave equations of KdV type (I), Chaos Soliton. Fract., 23 (2005), 469-475. doi: 10.1016/j.chaos.2004.04.027.  Google Scholar

[13]

S. Y. Lou and L. L. Chen, Solitary wave solutions and cnoidal wave solutions to the combined KdV and mKdV equation, Math. Meth. Appl. Sci., 17 (1994), 339-347. doi: 10.1002/mma.1670170503.  Google Scholar

[14]

W. Mafliet and W. Hereman, The tanh method: I . Exact solutions of nonlinear evolution and wave equations, Phys. Scr., 54 (1996), 563-568. doi: 10.1088/0031-8949/54/6/003.  Google Scholar

[15]

R. M. Miura, Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation, J. Math. Phys., 9 (1968), 1202-1204. doi: 10.1063/1.1664700.  Google Scholar

[16]

R. M. Miura, C. S. Gardner and M. D. Kruskal, Korteweg-de vries equation and generalizations. II. existence of conservation laws and constants of motion, J. Math. Phys., 9 (1968), 1204-1209. doi: 10.1063/1.1664701.  Google Scholar

[17]

A. C. Newell, Solitons in Mathematics and Physics, SIAM, Philadelphia, PA, 1985. doi: 10.1137/1.9781611970227.  Google Scholar

[18]

A. Saha, B. Talukdar and S. Chatterjee, Dynamical systems theory for the Gardner equation, Phys. Rev. E, 89 (2014), 023204. doi: 10.1103/PhysRevE.89.023204.  Google Scholar

[19]

M. Song, B. S. Ahmed, E. Zerrad and A. Biswas, Domain wall and bifurcation analysis of the Klein-Gordon Zakharov equation in (1 + 2)-dimensions with power law nonlinearity, Chaos, 23 (2013), 033115, 6pp. doi: 10.1063/1.4816346.  Google Scholar

[20]

M. Wadati, Wave propagation in nonlinear lattice. I, J. Phys. Soc. Jpn., 38 (1975), 673-680. doi: 10.1143/JPSJ.38.673.  Google Scholar

[21]

M. Wadati, Wave propagation in nonlinear lattice. II, J. Phys. Soc. Jpn., 38 (1975), 681-686. doi: 10.1143/JPSJ.38.673.  Google Scholar

[22]

A. M. Wazwaz, The extended tanh method for abundant solitary wave solutions of nonlinear wave equations, Appl. Math. Comput., 187 (2007), 1131-1142. doi: 10.1016/j.amc.2006.09.013.  Google Scholar

[23]

J. F. Zhang, New solitary wave solution of the combined KdV and mKdV equation, Int. J. Theor. Phys., 37 (1998), 1541-1546. doi: 10.1023/A:1026615919186.  Google Scholar

show all references

References:
[1]

G. Betchewe, K. K. Victor, B. B. Thomas and K. T. Crepin, New solutions of the Gardner equation: Analytical and numerical analysis of its dynamical understanding, Appl. Math. Comput., 223 (2013), 377-388. doi: 10.1016/j.amc.2013.08.028.  Google Scholar

[2]

A. Biswas and M. Song, Soliton solution and bifurcation analysis of the Zakharov-Kuznetsov-Benjamin-Bona-Mahoney equation with power law nonlinearity, Commun. Nonlinear Sci. Numer. Simulat., 18 (2013), 1676-1683. doi: 10.1016/j.cnsns.2012.11.014.  Google Scholar

[3]

Y. R. Chen and R. Liu, Some new nonlinear wave solutions for two (3+1)-dimensional equations, Appl. Math. Comput., 260 (2015), 397-411. doi: 10.1016/j.amc.2015.03.098.  Google Scholar

[4]

M. W. Coffey, On series expansions giving closed-form solutions of Korteweg-de Vries-like equations, SIAM J. Appl. Math., 50 (1990), 1580-1592. doi: 10.1137/0150093.  Google Scholar

[5]

Z. Fu, S. Liu and S. Liu, New kinds of solutions to Gardner equation, Chaos Soliton. Fract., 20 (2004), 301-309. doi: 10.1016/S0960-0779(03)00383-7.  Google Scholar

[6]

R. Grimshaw, D. Pelinovsky, E. Pelinovsky and A. Slunyaev, Generation of large-amplitude solitons in the extended Korteweg-de Vries equation, Chaos, 12 (2002), 1070-1076. doi: 10.1063/1.1521391.  Google Scholar

[7]

R. Grimshawa, D. Pelinovsky, E. Pelinovsky and T. Talipova, Wave group dynamics in weakly nonlinear long-wave models, Physica D, 159 (2001), 35-57. doi: 10.1016/S0167-2789(01)00333-5.  Google Scholar

[8]

R. Grimshaw, A. Slunyaev and E. Pelinovsky, Generation of solitons and breathers in the extended Korteweg-de Vries equation with positive cubic nonlinearity, Chaos, 20 (2010), 013102, 11pp. doi: 10.1063/1.3279480.  Google Scholar

[9]

K. Konno and Y. H. Ichikawa, A modified Korteweg de Vries equation for ion acoustic waves, J. Phys. Soc. Jpn., 37 (1974), 1631-1636. doi: 10.1143/JPSJ.37.1631.  Google Scholar

[10]

S. Liu, Z. Fu, S. Liu and Q. Zhao, Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Phys. Lett. A, 289 (2001), 69-74. doi: 10.1016/S0375-9601(01)00580-1.  Google Scholar

[11]

R. Liu and W. F. Yan, Some common expressions and new bifurcation phenomena for nonlinear waves in a generalized mKdV equation, Int. J. Bifurcat. Chaos, 23 (2013), 1330007, 19pp. doi: 10.1142/S0218127413300073.  Google Scholar

[12]

Y. Long, W. G. Rui and B. He, Travelling wave solutions for a higher order wave equations of KdV type (I), Chaos Soliton. Fract., 23 (2005), 469-475. doi: 10.1016/j.chaos.2004.04.027.  Google Scholar

[13]

S. Y. Lou and L. L. Chen, Solitary wave solutions and cnoidal wave solutions to the combined KdV and mKdV equation, Math. Meth. Appl. Sci., 17 (1994), 339-347. doi: 10.1002/mma.1670170503.  Google Scholar

[14]

W. Mafliet and W. Hereman, The tanh method: I . Exact solutions of nonlinear evolution and wave equations, Phys. Scr., 54 (1996), 563-568. doi: 10.1088/0031-8949/54/6/003.  Google Scholar

[15]

R. M. Miura, Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation, J. Math. Phys., 9 (1968), 1202-1204. doi: 10.1063/1.1664700.  Google Scholar

[16]

R. M. Miura, C. S. Gardner and M. D. Kruskal, Korteweg-de vries equation and generalizations. II. existence of conservation laws and constants of motion, J. Math. Phys., 9 (1968), 1204-1209. doi: 10.1063/1.1664701.  Google Scholar

[17]

A. C. Newell, Solitons in Mathematics and Physics, SIAM, Philadelphia, PA, 1985. doi: 10.1137/1.9781611970227.  Google Scholar

[18]

A. Saha, B. Talukdar and S. Chatterjee, Dynamical systems theory for the Gardner equation, Phys. Rev. E, 89 (2014), 023204. doi: 10.1103/PhysRevE.89.023204.  Google Scholar

[19]

M. Song, B. S. Ahmed, E. Zerrad and A. Biswas, Domain wall and bifurcation analysis of the Klein-Gordon Zakharov equation in (1 + 2)-dimensions with power law nonlinearity, Chaos, 23 (2013), 033115, 6pp. doi: 10.1063/1.4816346.  Google Scholar

[20]

M. Wadati, Wave propagation in nonlinear lattice. I, J. Phys. Soc. Jpn., 38 (1975), 673-680. doi: 10.1143/JPSJ.38.673.  Google Scholar

[21]

M. Wadati, Wave propagation in nonlinear lattice. II, J. Phys. Soc. Jpn., 38 (1975), 681-686. doi: 10.1143/JPSJ.38.673.  Google Scholar

[22]

A. M. Wazwaz, The extended tanh method for abundant solitary wave solutions of nonlinear wave equations, Appl. Math. Comput., 187 (2007), 1131-1142. doi: 10.1016/j.amc.2006.09.013.  Google Scholar

[23]

J. F. Zhang, New solitary wave solution of the combined KdV and mKdV equation, Int. J. Theor. Phys., 37 (1998), 1541-1546. doi: 10.1023/A:1026615919186.  Google Scholar

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