Travelling wave solutions to Zufiria’s higher-order Boussinesq type equations. (English) Zbl 1238.35131
Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71, No. 12, e-Suppl., e711-e724 (2009).
Summary: Zufiria’s higher-order Boussinesq type equations are studied by transforming them into solvable ordinary differential equations. Various families of their travelling wave solutions are generated, which include periodic wave, solitary wave, periodic-like wave, soliton-like wave, Jacobi elliptic function periodic wave, combined non-degenerative Jacobi elliptic function-like wave, Weierstrass elliptic function periodic and rational solutions. The presented approach can be also applied to nonlinear wave equations with variable coefficients in mathematical physics and mechanics.
MSC:
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 35Q51 | Soliton equations |
| 35C07 | Traveling wave solutions |
| 35A24 | Methods of ordinary differential equations applied to PDEs |
Keywords:
Zufiria’s higher-order Boussinesq type equations; periodic wave; solitary wave; Jacobi elliptic function; Weierstrass elliptic function; rational solutionsSoftware:
RATHReferences:
| [1] | Ablowitz, M. J.; Clarkson, P. A., Soliton, Nonlinear Evolution Equations and Inverse Scattering (1991), Cambridge University press: Cambridge University press Cambridge · Zbl 0762.35001 |
| [2] | Wadati, M., J. Phys. Soc. Japan, 53, 2642 (1983) |
| [3] | Lamb, G. L., Elements of Soliton Theory (1980), Wiley: Wiley New York · Zbl 0445.35001 |
| [4] | Wadati, M.; Sanuki, H.; Konno, K., Prog. Theor. Phys., 53, 419 (1975) · Zbl 1079.35506 |
| [5] | Konno, K.; Wadati, M., Prog. Theor. Phys., 53, 1652 (1975) · Zbl 1079.35505 |
| [6] | Matveev, V. B.; Salle, M. A., Darboux Transformations and Solitons (1991), Springer: Springer Berlin · Zbl 0744.35045 |
| [7] | Hirota, R., J. Math. Phys., 14, 810 (1973) · Zbl 0261.76008 |
| [8] | Cariello, F.; Tabor, M., Phys. D, 39, 77 (1989) · Zbl 0687.35093 |
| [9] | Ma, W. X.; Fuchssteiner, B., Internat. J. Non-linear Mech., 31, 3, 329 (1996) · Zbl 0863.35106 |
| [10] | Parkes, E. J.; Duffy, B. R., Comput. Phys. Commun., 98, 288 (1996) · Zbl 0948.76595 |
| [11] | Li, Z. B.; Liu, Y. P., Comput. Phys. Commun., 148, 256 (2002) · Zbl 1196.35008 |
| [12] | Wazwaz, A. M., Chaos Solitons Fractals, 28, 1005 (2006) · Zbl 1099.35125 |
| [13] | Liu, S. K.; Fu, Z. T.; Liu, S. D., Phys. Lett. A, 351, 59 (2006) · Zbl 1234.35227 |
| [14] | Yan, Z. Y., Chaos Solitons Fractals, 18, 299 (2003) · Zbl 1069.37060 |
| [15] | Chen, Y.; Wang, Q., Chinese Phys., 13, 1796 (2004) |
| [16] | Zhu, J. M.; Ma, Z. Y.; Fang, J. P.; Zheng, C. L.; Zhang, J. F., Chinese Phys., 13, 798 (2004) |
| [17] | Li, J. B.; Zhang, L. J., Chaos Solitons Fractals, 14, 581 (2002) · Zbl 0997.35096 |
| [18] | Zhou, Y. B.; Wang, M. L.; Wang, Y. M., Phys. Lett. A, 308, 31 (2003) · Zbl 1008.35061 |
| [19] | Wang, M. L.; Li, X. Z., Chaos Solitons Fractals, 24, 1257 (2005) · Zbl 1092.37054 |
| [20] | Li, X. Z.; Zhang, J. L.; Wang, Y. M., Acta Phys. Sinica, 53, 4045 (2004) · Zbl 1202.34030 |
| [21] | Li, X. Y.; Yang, S.; Wang, M. L., Chaos Soitons Fractals, 25, 629 (2005) · Zbl 1068.35120 |
| [22] | Zhang, J. L.; Wang, M. L.; Wang, Y. M.; Fang, Z. D., Phys. Lett. A, 350, 103 (2006) · Zbl 1195.65211 |
| [23] | Salah, M. E.; Doğan, K., Appl. Math. Comput., 157, 93 (2004) · Zbl 1061.65100 |
| [24] | Sirendaoreji, Phys. Lett. A, 356, 124 (2006) · Zbl 1160.35527 |
| [25] | Ma, W. X.; Wu, H. Y.; He, J. S., Phys. Lett. A, 364, 29 (2007) · Zbl 1203.35059 |
| [26] | Xu, G. Q.; Li, Z. B., Chaos Solitons Fractals, 24, 549 (2004) |
| [27] | Xu, C. Z.; He, B. G.; Zhang, J. F., Chinese Phys., 15, 1 (2006) |
| [28] | Ma, W. X.; You, Y. C., Trans. Amer. Math. Soc., 357, 6, 1753 (2004) |
| [29] | Hon, Y. C.; Fan, E. G., Chaos Solitons Fractals, 19, 1141 (2004) · Zbl 1068.35130 |
| [30] | Yan, Z. Y., Chaos Solitons Fractals, 21, 1013 (2004) · Zbl 1046.35103 |
| [31] | Zhi, H. Y.; Wang, Q.; Zhang, H. Q., Acta Phys. Sinica, 54, 1002 (2005) · Zbl 1202.35291 |
| [32] | Zeng, X.; Zhang, H. Q., Acta Phys. Sinica, 54, 504 (2005) · Zbl 1202.35280 |
| [33] | Bai, C. L.; Zhao, H., Phys. Lett. A, 354, 428 (2006) |
| [34] | Yomba, E., Chaos Solitons Fractals, 27, 187 (2006) · Zbl 1088.35532 |
| [35] | Zufiria, J. A., J. Fluid Mech., 180, 371 (1987) · Zbl 0622.76017 |
| [36] | Bridges, T. J.; Fan, E. G., Phys. Lett. A, 326, 381 (2004) · Zbl 1138.76324 |
| [37] | Byrd, P. F.; Friedman, M. D., Handbook of Elliptic Integrals for Engineers and Scientists (1971), Springer-Verlag Berlin Heidelberg: Springer-Verlag Berlin Heidelberg New York · Zbl 0213.16602 |
| [38] | Patrick, D. V., Elliptic Function and Elliptic Curves (1973), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0261.33001 |
| [39] | Chamdrasekharan, K., Elliptic Functions (1978), Springer: Springer Berlin |
| [40] | Andrews, G. E.; Askey, R.; Roy, R., Special Functions (2004), Tsinghua University Press: Tsinghua University Press Beijing |
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