SUB-MANIFOLD AND TRAVELING WAVE SOLUTIONS OF ITO'S 5TH-ORDER MKDV EQUATION

Lijun Zhang; Haixia Chang; Chaudry Masood Khalique

doi:10.11948/2017086

2017 Volume 7 Issue 4

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Lijun Zhang, Haixia Chang, Chaudry Masood Khalique. SUB-MANIFOLD AND TRAVELING WAVE SOLUTIONS OF ITO'S 5TH-ORDER MKDV EQUATION[J]. Journal of Applied Analysis & Computation, 2017, 7(4): 1417-1430. doi: 10.11948/2017086

Citation:

Lijun Zhang, Haixia Chang, Chaudry Masood Khalique. SUB-MANIFOLD AND TRAVELING WAVE SOLUTIONS OF ITO'S 5TH-ORDER MKDV EQUATION[J]. Journal of Applied Analysis & Computation, 2017, 7(4): 1417-1430. doi: 10.11948/2017086

SUB-MANIFOLD AND TRAVELING WAVE SOLUTIONS OF ITO'S 5TH-ORDER MKDV EQUATION

1 Department of Mathematics, School of Science, Zhejiang Sci-Tech University, Hangzhou, Zhejiang, 310018, China;
2 International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X 2046, Mmabatho, 2735, South Africa

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Abstract

Abstract

In this paper, we study Ito's 5th-order mKdV equation with the aid of symbolic computation system and by qualitative analysis of planar dynamical systems. We show that the corresponding higher-order ordinary differential equation of Ito's 5th-order mKdV equation, for some particular values of the parameter, possesses some sub-manifolds defined by planar dynamical systems. Some solitary wave solutions, kink and periodic wave solutions of the Ito's 5th-order mKdV equation for these particular values of the parameter are obtained by studying the bifurcation and solutions of the corresponding planar dynamical systems.
- Ito's 5th-order mKdV equation /
- traveling wave solutions /
- subequations /
- planar dynamical systems
MSC: 34C25;34C37;34C45

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References

[1]	A. Chen, S. Wen and W. Huang, Existence and orbital stability of periodic wave solutions for the nonlinear Schodinger equation, J. Appl. Anal. Comp., 2012, 2(2), 137-148. Google Scholar
[2]	A. Chen, S. Wen, S. Tang, W. Huang and Z. Qiao, Effects of quadratic singular curves in integrable equations, Stud. Appl. Math., 2015, 134, 24-61. Google Scholar
[3]	S. N. Chow and J. K. Hale, Method of Bifurcation Theory, Springer, New York, 1981. Google Scholar
[4]	H. Ding and L-Q Chen, Galerkin methods for natural frequencies of high-speed axially moving beams, J. Sound Vib., 2010, 329(17), 3484-3494. Google Scholar
[5]	H. Ding, L-Q Chen and S. P. Yang, Convergence of Galerkin truncation for dynamic response of finite beams on nonlinear foundations under a moving load, J. Sound Vib., 2012, 331(10), 2426-2442. Google Scholar
[6]	H. H Dai and X. H. Zhao, Nonlinear traveling waves in a rod composed of a modified Mooney-Rivlin material. I Bifurcation of critical points and the nonsingular case, Proc. R. Soc. Lond. A, 1999, 455, 3845-3874. Google Scholar
[7]	E. Fan, Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A, 2000, 277, 212-218. Google Scholar
[8]	I. S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products, Sixth Edition, Academic Press, New York, 2000. Google Scholar
[9]	J. H. He and X. H. Wu, Exp-function method for nonlinear wave equations, Chaos Solitons Fractals, 2006, 30(3), 700-708. Google Scholar
[10]	M. Ito, An extension of nonlinear evolution equation of KdV (mKdV) type to higher orders, J. Phys. Soc. Jpn., 1980, 49, 771-778. Google Scholar
[11]	J. B. Li, Singular Traveling Wave Equations:Bifurcations and Exact Solutions. Science Press, Beijing, 2013. Google Scholar
[12]	W. X. Ma and J. H. Lee, A transformed rational function method and exact solutions to the 3+1 dimensional Jimbo-Miwa equation, Chaos Solitons Fractals, 2009, 42, 1356-1363. Google Scholar
[13]	E. J. Parkes and B. R. Duffy, An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations, Comp. Phys. Commun., 1996, 98, 288-300. Google Scholar
[14]	E. J. Parkes, B. R. Duffy and P. C. Abbott, The Jacobi elliptic-function method for finding periodic-wave solutions to nonlinear evolution equations, Phys. Lett. A, 2002, 295, 280-286. Google Scholar
[15]	W. G. Rui, Different kinds of exact solutions with two-loop character of the two-component short pulse equations of the first kind, Commun. Nonlinear Sci. Numer. Simulat., 2013, 18, 2667-2678. Google Scholar
[16]	S. Shen and Z. Pan, A note on the Jacobi elliptic function expansion method, Phys. Lett. A, 2003, 308, 143-148. Google Scholar
[17]	J. Shen, Shock wave solutions of the compound Burgers-Korteweg-de equation, Appl. Math. Comp., 2008, 96(2), 842-849. Google Scholar
[18]	J. Shen, B. Miao and J. Luo, Bifurcations and Highly Nonlinear Traveling Waves in Periodic Dimer Granular Chains, Math. Method Appl. Sci., 2011, 34(12), 1445-1449. Google Scholar
[19]	A. M. Wazwaz, A sine-cosine method for handling nonlinear wave equations, Math. Comput Model, 2004, 40(5-6), 499-508. Google Scholar
[20]	Y. Zhang, S. Lai, J. Yin and Y. Wu, The application of the auxiliary equation technique to a generalized mKdV equation with variable coefficients, J. Comput. Appl. Math., 2009, 223, 75-85. Google Scholar
[21]	L. J. Zhang and C. M. Khalique, Exact solitary wave and quasi-periodic wave solutions of the KdV-Sawada-Kotera-Ramani equation, Adv. Differ. Equ., 2015, 195. Google Scholar
[22]	L. J. Zhang and C. M. Khalique, Exact solitary wave and periodic wave solutions of the Kaup-Kuper-Schmidt equation, J. Appl. Anal. Comput., 2015, 5(3), 485-495. Google Scholar