# American Institute of Mathematical Sciences

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

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

 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.
Citation: Yiren Chen, Zhengrong Liu. The bifurcations of solitary and kink waves described by the Gardner equation. Discrete and 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [17] A. C. Newell, Solitons in Mathematics and Physics, SIAM, Philadelphia, PA, 1985. doi: 10.1137/1.9781611970227. [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. [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. [20] M. Wadati, Wave propagation in nonlinear lattice. I, J. Phys. Soc. Jpn., 38 (1975), 673-680. doi: 10.1143/JPSJ.38.673. [21] M. Wadati, Wave propagation in nonlinear lattice. II, J. Phys. Soc. Jpn., 38 (1975), 681-686. doi: 10.1143/JPSJ.38.673. [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. [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.

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##### 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [17] A. C. Newell, Solitons in Mathematics and Physics, SIAM, Philadelphia, PA, 1985. doi: 10.1137/1.9781611970227. [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. [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. [20] M. Wadati, Wave propagation in nonlinear lattice. I, J. Phys. Soc. Jpn., 38 (1975), 673-680. doi: 10.1143/JPSJ.38.673. [21] M. Wadati, Wave propagation in nonlinear lattice. II, J. Phys. Soc. Jpn., 38 (1975), 681-686. doi: 10.1143/JPSJ.38.673. [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. [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.
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