American Institute of Mathematical Sciences

Advanced
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 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.

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.

[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.

[1]

Jerry L. Bona, Angel Durán, Dimitrios Mitsotakis. Solitary-wave solutions of Benjamin-Ono and other systems for internal waves. I. approximations. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 87-111. doi: 10.3934/dcds.2020215
[2]

Anwar Ja'afar Mohamad Jawad, Mohammad Mirzazadeh, Anjan Biswas. Dynamics of shallow water waves with Gardner-Kadomtsev-Petviashvili equation. Discrete and Continuous Dynamical Systems - S, 2015, 8 (6) : 1155-1164. doi: 10.3934/dcdss.2015.8.1155
[3]

Claudio Muñoz. The Gardner equation and the stability of multi-kink solutions of the mKdV equation. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3811-3843. doi: 10.3934/dcds.2016.36.3811
[4]

H. Kalisch. Stability of solitary waves for a nonlinearly dispersive equation. Discrete and Continuous Dynamical Systems, 2004, 10 (3) : 709-717. doi: 10.3934/dcds.2004.10.709
[5]

Amin Esfahani, Steve Levandosky. Solitary waves of the rotation-generalized Benjamin-Ono equation. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 663-700. doi: 10.3934/dcds.2013.33.663
[6]

Steve Levandosky, Yue Liu. Stability and weak rotation limit of solitary waves of the Ostrovsky equation. Discrete and Continuous Dynamical Systems - B, 2007, 7 (4) : 793-806. doi: 10.3934/dcdsb.2007.7.793
[7]

Khaled El Dika. Asymptotic stability of solitary waves for the Benjamin-Bona-Mahony equation. Discrete and Continuous Dynamical Systems, 2005, 13 (3) : 583-622. doi: 10.3934/dcds.2005.13.583
[8]

Sevdzhan Hakkaev. Orbital stability of solitary waves of the Schrödinger-Boussinesq equation. Communications on Pure and Applied Analysis, 2007, 6 (4) : 1043-1050. doi: 10.3934/cpaa.2007.6.1043
[9]

Jundong Wang, Lijun Zhang, Elena Shchepakina, Vladimir Sobolev. Solitary waves of singularly perturbed generalized KdV equation with high order nonlinearity. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022124
[10]

Jerry L. Bona, Thierry Colin, Colette Guillopé. Propagation of long-crested water waves. Ⅱ. Bore propagation. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5543-5569. doi: 10.3934/dcds.2019244
[11]

Zengji Du, Xiaojie Lin, Yulin Ren. Dynamics of solitary waves and periodic waves for a generalized KP-MEW-Burgers equation with damping. Communications on Pure and Applied Analysis, 2022, 21 (6) : 1987-2003. doi: 10.3934/cpaa.2021118
[12]

José Raúl Quintero, Juan Carlos Muñoz Grajales. Solitary waves for an internal wave model. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5721-5741. doi: 10.3934/dcds.2016051
[13]

Jerry Bona, Hongqiu Chen. Solitary waves in nonlinear dispersive systems. Discrete and Continuous Dynamical Systems - B, 2002, 2 (3) : 313-378. doi: 10.3934/dcdsb.2002.2.313
[14]

Orlando Lopes. A linearized instability result for solitary waves. Discrete and Continuous Dynamical Systems, 2002, 8 (1) : 115-119. doi: 10.3934/dcds.2002.8.115
[15]

Xingxing Liu. Stability in the energy space of the sum of $ N $ solitary waves for an equation modelling shallow water waves of moderate amplitude. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022105
[16]

John Boyd. Strongly nonlinear perturbation theory for solitary waves and bions. Evolution Equations and Control Theory, 2019, 8 (1) : 1-29. doi: 10.3934/eect.2019001
[17]

Emmanuel Hebey. Solitary waves in critical Abelian gauge theories. Discrete and Continuous Dynamical Systems, 2012, 32 (5) : 1747-1761. doi: 10.3934/dcds.2012.32.1747
[18]

Ola I. H. Maehlen. Solitary waves for weakly dispersive equations with inhomogeneous nonlinearities. Discrete and Continuous Dynamical Systems, 2020, 40 (7) : 4113-4130. doi: 10.3934/dcds.2020174
[19]

Yohei Yamazaki. Center stable manifolds around line solitary waves of the Zakharov–Kuznetsov equation with critical speed. Discrete and Continuous Dynamical Systems, 2021, 41 (8) : 3579-3614. doi: 10.3934/dcds.2021008
[20]

Yuanhong Wei, Yong Li, Xue Yang. On concentration of semi-classical solitary waves for a generalized Kadomtsev-Petviashvili equation. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 1095-1106. doi: 10.3934/dcdss.2017059

2021 Impact Factor: 1.865

Tools

Metrics

  • PDF downloads (150)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy