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Sharp variational characterization and a Schrödinger equation with Hartree type nonlinearity
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 |
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. |
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