×

The \(K(m, n)\) equation with generalized evolution term studied by symmetry reductions and qualitative analysis. (English) Zbl 1246.35172

Summary: The authors obtain symmetry reductions of the \(K(m, n)\) equation with generalized evolution term. The reduction to ordinary differential equations comes from an optimal system of subalgebras. Some of these equations admit symmetries which lead to further reductions, and one of them comes out suitable for qualitative analysis. Its dynamical behavior is fully described and conservative quantities are stated.

MSC:

35Q51 Soliton equations
35B06 Symmetries, invariants, etc. in context of PDEs
35A24 Methods of ordinary differential equations applied to PDEs
34A05 Explicit solutions, first integrals of ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Biswas, A., 1-soliton solution of the \(K(m,n)\) equation with generalized evolution, Phys. Lett. A, 372, 4601-4602 (2008) · Zbl 1221.35099
[2] Bruzón, M. S.; Gandarias, M. L., Travelling wave solutions of the \(K(m,n)\) equation with generalized evolution, Math. Methods Appl. Sci. (2010) · Zbl 1221.35024
[3] Chen, A.; Li, J., Single peak solitary wave solutions for the osmosis \(K(2, 2)\) equation under inhomogeneous boundary condition, J. Math. Anal. Appl., 369, 758-766 (2010) · Zbl 1196.35180
[4] Liu, H.; Li, J., Lie symmetry analysis and exact solutions for the extended mKdV equation, Acta Appl. Math., 109, 1107-1119 (2008) · Zbl 1223.37079
[5] Rosenau, P.; Hyman, J. M., Compactons: solitons with finite wavelengths, Phys. Rev. Lett., 70, 5, 564-567 (1993) · Zbl 0952.35502
[6] Rosenau, P., On nonanalytic solitary waves formed by a nonlinear dispersion, Phys. Lett. A, 230, 305-318 (1997) · Zbl 1052.35511
[7] Rosenau, P., Nonlinear dispersion and compact structures, Phys. Rev. Lett., 73, 13, 1737-1741 (1994) · Zbl 0953.35501
[8] Rosenau, P., On a class of nonlinear dispersive-dissipative interactions, Physica D, 230, 5-6, 535-546 (1998) · Zbl 0938.35172
[9] Qu, C. Z.; Zhang, S.; Zhang, Q., Integrability of models arising from motions of plane curves, Z. Naturforsch., 58a, 75-83 (2003)
[10] Chou, K. S.; Qu, C. Z., Integrable equations arising from motions of plane curves. II, J. Nonlinear Sci., 13, 487-517 (2003) · Zbl 1045.35063
[11] Tang, S.; Huang, W., Bifurcations of travelling wave solutions for the \(K(n, - n, 2 n)\) equations, Appl. Math. Comput., 203, 39-49 (2008) · Zbl 1157.37021
[12] Tang, S.; Li, M., Bifurcations of travelling wave solutions in a class of generalized KdV equation, Appl. Math. Comput., 177, 589-596 (2006) · Zbl 1099.35123
[13] Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions (1972), Dover: Dover New York · Zbl 0515.33001
[14] Olver, P., Applications of Lie Groups to Differential Equations (1993), Springer-Verlag: Springer-Verlag New York
[15] Ma, W. X.; Chen, M., Do symmetry constraints yield exact solutions?, Chaos Solitons Fract., 32, 1513-1517 (2007) · Zbl 1137.35325
[16] Ma, W. X.; Wu, H., Time-space integrable decompositions of nonlinear evolution equations, J. Math. Anal. Appl., 324, 134-149 (2006) · Zbl 1109.35098
[17] Ma, W. X.; Wu, H.; He, J., Partial differential equations possesing Frobenius integrable decompositions, Phys. Lett. A, 364, 29-32 (2007) · Zbl 1203.35059
[18] Perko, L., Differential Equations and Dynamical Systems (1996), Springer-Verlag · Zbl 0854.34001
[19] Strogatz, S. H., Nonlinear Dynamics and Chaos (1994), Perseus Publishing: Perseus Publishing Cambridge Massachusetts
[20] Wiggins, S., Introduction to Applied Nonlinear Dynamical Systems and Chaos (2003), Springer-Verlag · Zbl 1027.37002
[21] Romero, J. L.; Gandarias, M. L.; Medina, E., Symmetries, periodic plane waves and blow-up of \(\lambda-\omega\) systems, Physica D, 147, 259-272 (2000) · Zbl 0963.35085
[22] Camacho, V.; Guy, R. D.; Jacobsen, J., Travelling waves and shocks in a viscoelastic generalization of Burgers’ equation, SIAM J. Appl. Math., 68, 5 (2008) · Zbl 1152.76010
[23] Tang, S.; Huang, X.; Huang, W., Bifurcations of travelling wave solutions for the generalized KP-BBM equation, Appl. Math. Comput., 216, 2881-2890 (2010) · Zbl 1198.37105
[24] Xie, Y.; Zhou, B.; Tang, S., Bifurcations of travelling wave solutions for the generalized (2+1)-dimensional Boussinesq-Kadomtsev-Petviashvili equation, Appl. Math. Comput, 217, 6, 2433-2447 (2010) · Zbl 1202.35270
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.