# American Institute of Mathematical Sciences

August  2018, 11(4): 759-772. doi: 10.3934/dcdss.2018048

## Classification and bifurcation of a class of second-order ODEs and its application to nonlinear PDEs

 a. Department of Mathematics, School of Science, Zhejiang Sci-Tech University, Hangzhou, Zhejiang 310018, China b. International Institute for Symmetry Analysis and Mathematical Modeling, North-West University, Mafikeng Campus, P Bag X2046, Mafikeng, South Africa c. College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, Shandong 266590, China

* Corresponding author: Lijun Zhang

Received  December 2016 Published  November 2017

In this paper, by using dynamical system theorems we study the bifurcation of a second-order ordinary differential equation which can be obtained from many nonlinear partial differential equations via traveling wave transformation and integrations. We present all the bounded exact solutions of this second-order ordinary differential equation which contains four parameters by normalization and classification. As a result, one can obtain all possible bounded exact traveling wave solutions including soliatry waves, kink and periodic wave solutions of many nonlinear wave equations by the formulas presented in this paper. As an example, all bounded traveling wave solutions of the modified regularized long wave equation are obtained to illustrate our approach.

Citation: Lijun Zhang, Chaudry Masood Khalique. Classification and bifurcation of a class of second-order ODEs and its application to nonlinear PDEs. Discrete and Continuous Dynamical Systems - S, 2018, 11 (4) : 759-772. doi: 10.3934/dcdss.2018048
##### References:
 [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.  doi: 10.1016/j.cnsns.2008.03.014. [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.  doi: 10.1098/rsta.1972.0032. [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.  doi: 10.1111/1467-9590.00163. [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. doi: 10.1007/978-1-4612-1140-2. [7] D. J. Kaup, A higher order water wave equation and method for solving it, Progr. Theor. Phys., 54 (1976), 396-408.  doi: 10.1143/PTP.54.396. [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.  doi: 10.1016/j.amc.2014.12.094. [9] S. Lai and X. Lv, The Jacobi elliptic function solutions to a generalized Benjamin-Bona-Mahony equation, Mathematical and Computer Modelling, 49 (2009), 369-378.  doi: 10.1016/j.mcm.2008.03.009. [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.  doi: 10.1142/S0218127411028581. [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.  doi: 10.1016/S0960-0779(01)00248-X. [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.  doi: 10.1016/j.chaos.2009.03.098. [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.  doi: 10.1016/0167-2789(88)90095-4. [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.  doi: 10.1016/j.cam.2007.12.021. [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. doi: 10.1186/s13662-015-0510-y. [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. doi: 10.1155/2015/548606.

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##### References:
 [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.  doi: 10.1016/j.cnsns.2008.03.014. [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.  doi: 10.1098/rsta.1972.0032. [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.  doi: 10.1111/1467-9590.00163. [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. doi: 10.1007/978-1-4612-1140-2. [7] D. J. Kaup, A higher order water wave equation and method for solving it, Progr. Theor. Phys., 54 (1976), 396-408.  doi: 10.1143/PTP.54.396. [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.  doi: 10.1016/j.amc.2014.12.094. [9] S. Lai and X. Lv, The Jacobi elliptic function solutions to a generalized Benjamin-Bona-Mahony equation, Mathematical and Computer Modelling, 49 (2009), 369-378.  doi: 10.1016/j.mcm.2008.03.009. [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.  doi: 10.1142/S0218127411028581. [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.  doi: 10.1016/S0960-0779(01)00248-X. [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.  doi: 10.1016/j.chaos.2009.03.098. [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.  doi: 10.1016/0167-2789(88)90095-4. [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.  doi: 10.1016/j.cam.2007.12.021. [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. doi: 10.1186/s13662-015-0510-y. [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. doi: 10.1155/2015/548606.
The phase portrait of system (1.3) with $a_3=1, a_2=1, a_0=0$ and (a) $a_1=-1$; (b) $a_1=\frac{2}{9}$; (c) $a_1=\frac{1}{5}$; (d) $a_1=\frac{17}{72}$.
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