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Discrete and Continuous Dynamical Systems - Series S (DCDS-S)
 

Classification and bifurcation of a class of second-order ODEs and its application to nonlinear PDEs
Pages: 759 - 772, Issue 4, August 2018

doi:10.3934/dcdss.2018048      Abstract        References        Full text (412.8K)           Related Articles

Lijun Zhang - Department of Mathematics, School of Science, Zhejiang Sci-Tech University, Hangzhou, Zhejiang 310018, China (email)
Chaudry Masood Khalique - International Institute for Symmetry Analysis and Mathematical Modeling, North-West University, Mafikeng Campus, P Bag X2046, Mafikeng, South Korea (email)

1 A. Bekir, On traveling wave solutions to combined KdV-MKdV equation and modified Burgers-KdV equation, Commun Nonlinear Sci Numer Simulat., 14 (2009), 1038-1042.       
2 T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equation for long waves in nonlinear dispersive system, Philos. Trans. Royal. Soc. Lond. Ser. A, 272 (1972), 47-78.       
3 S. N. Chow and J. K. Hale, Method of Bifurcation Theory, Springer-Verlag, New York-Berlin, 1982.       
4 G. A. El, R. H. J. Grinshaw and M. V. Paclov, Integrable shallow-water equations and undular bores, Studies in Applied Mathematics, 106 (2001), 157-186.       
5 I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, $6^{th}$ edition, Academic Press, New York, 2000.
6 J. Guckenheimer and P. Holmes, Dynamical Systems and Bifurcations of Vector Fields, Springer, New York, 1983.       
7 D. J. Kaup, A higher order water wave equation and method for solving it, Progr. Theor. Phys., 54 (1976), 396-408.       
8 B. Kilic and M. Inc, The first integral method for the time fractional Kaup-Boussinesq system with time dependent coefficient, Applied Mathematics and Computation, 254 (2015), 70-74.       
9 S. Lai, X. Lv and M Shuai, The Jacobi elliptic function solutions to a generalized Benjamin-Bona-Mahony equation, Mathematical and Computer Modelling, 49 (2009), 369-378.       
10 J. B. Li and Y. Zhang, Homoclinic manifolds, center manifolds and exact solutions of four-dimensional traveling wave systems for two classes of nonlinear wave equations, Int. J. Bifurcation and Chaos, 21 (2011), 527-543.       
11 J. B. Li and L. J. Zhang, Bifurcations of traveling wave solutions in generalized Pochhammer-Chree equation, Chaos, Solitons & Fractals, 14 (2002), 581-593.       
12 J. B. Li and Singular, Traveling Wave Equations: Bifurcations and Exact Solutions, Science Press, Beijing, 2013.
13 K. R. Raslan, Numerical study of the Modified Regularized Long Wave (MRLW) equation, Chaos, Solitons and Fractals, 42 (2009), 1845-1853.       
14 S. L. Robert, On the integrable variant of the Boussinesq system: Painlev property, rational solutions, a related many-body system, and equivalence with the AKNS hierarchy, Physica D: Nonlinear Phenomena, 30 (1988), 1-27.       
15 Y. Zhang, S. Lai, J. Yin and Y. Wu, The application of the auxiliary equation technique to a generalized mKdV equation with variable coefficients, Journal of Computational and Applied Mathematics, 223 (2009), 75-85.       
16 L. J. Zhang and C. M. Khalique, Exact solitary wave and periodic wave solutions of the Kaup-Kuper-Schmidt equation, Journal of Applied Analysis and Computation, 5 (2015), 485-495.       
17 L. J. Zhang and C. M. Khalique, Exact solitary wave and quasi-periodic wave solutions of the KdV-Sawada-Kotera-Ramani equation, Advances in Difference Equations, 2015 (2015), 12pp.       
18 L. J. Zhang and C. M. Khalique, Exact Solitary wave and periodic wave solutions of a class of higher-order nonlinear wave equations, Mathematical Problems in Engineering, 2015 (2015), Art. ID 548606, 8 pp.       

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