##
**Exact traveling wave solutions of explicit type, implicit type, and parametric type for \(K(m, n)\) equation.**
*(English)*
Zbl 1251.35138

Summary: By using the integral bifurcation method, we study the nonlinear \(K(m, n)\) equation for all possible values of \(m\) and \(n\). Some new exact traveling wave solutions of explicit type, implicit type, and parametric type are obtained. These exact solutions include peculiar compacton solutions, singular periodic wave solutions, compacton-like periodic wave solutions, periodic blowup solutions, smooth soliton solutions, and kink and antikink wave solutions. The great parts of them are different from the results in existing references. In order to show their dynamic profiles intuitively, the solutions of \(K(n, n)\), \(K(2n - 1, n)\), \(K(3n - 2, n)\), \(K(4n - 3, n)\), and \(K(m, 1)\) equations are chosen to illustrate with the concrete features.

### MSC:

35Q53 | KdV equations (Korteweg-de Vries equations) |

35C08 | Soliton solutions |

35C07 | Traveling wave solutions |

PDF
BibTeX
XML
Cite

\textit{X. Wu} et al., J. Appl. Math. 2012, Article ID 236875, 23 p. (2012; Zbl 1251.35138)

Full Text:
DOI

### References:

[1] | P. Rosenau and J. M. Hyman, “Compactons: solitons with finite wavelength,” Physical Review Letters, vol. 70, no. 5, pp. 564-567, 1993. · Zbl 0952.35502 |

[2] | P. Rosenau, “Nonlinear dispersion and compact structures,” Physical Review Letters, vol. 73, no. 13, pp. 1737-1741, 1994. · Zbl 0953.35501 |

[3] | P. Rosenau, “On solitons, compactons, and lagrange maps,” Physics Letters. A, vol. 211, no. 5, pp. 265-275, 1996. · Zbl 1059.35524 |

[4] | P. J. Olver and P. Rosenau, “Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support,” Physical Review E, vol. 53, no. 2, pp. 1900-1906, 1996. |

[5] | P. Rosenau, “On nonanalytic solitary waves formed by a nonlinear dispersion,” Physics Letters. A, vol. 230, no. 5-6, pp. 305-318, 1997. · Zbl 1052.35511 |

[6] | P. Rosenau, “On a class of nonlinear dispersive-dissipative interactions,” Physica D, vol. 123, no. 1-4, pp. 525-546, 1998, Nonlinear waves and solitons in physical systems (Los Alamos, NM, 1997). · Zbl 0938.35172 |

[7] | A. M. Wazwaz, “New solitary-wave special solutions with compact support for the nonlinear dispersive K(m,n) equations,” Chaos, Solitons and Fractals, vol. 13, no. 2, pp. 321-330, 2002. · Zbl 1028.35131 |

[8] | A. M. Wazwaz, “Exact special solutions with solitary patterns for the nonlinear dispersive K(m,n) equations,” Chaos, Solitons and Fractals, vol. 13, no. 1, pp. 161-170, 2002. · Zbl 1027.35115 |

[9] | Y. Zhu and Z. Lü, “New exact solitary-wave special solutions for the nonlinear dispersive K(m,n) equations,” Chaos, Solitons and Fractals, vol. 27, no. 3, pp. 836-842, 2006. · Zbl 1088.35548 |

[10] | G. Domairry, M. Ahangari, and M. Jamshidi, “Exact and analytical solution for nonlinear dispersive K(m,p) equations using homotopy perturbation method,” Physics Letters. A, vol. 368, no. 3-4, pp. 266-270, 2007. · Zbl 1209.65108 |

[11] | L. X. Tian and J. L. Yin, “Shock-peakon and shock-compacton solutions for K(p,q) equation by variational iteration method,” Journal of Computational and Applied Mathematics, vol. 207, no. 1, pp. 46-52, 2007. · Zbl 1119.65099 |

[12] | A. M. Wazwaz, “A reliable treatment of the physical structure for the nonlinear equation K(m,n),” Applied Mathematics and Computation, vol. 163, no. 3, pp. 1081-1095, 2005. · Zbl 1072.35580 |

[13] | Y. Zhu and X. Gao, “Exact special solitary solutions with compact support for the nonlinear dispersive K(m,n) equations,” Chaos, Solitons and Fractals, vol. 27, no. 2, pp. 487-493, 2006. · Zbl 1088.35547 |

[14] | Y. Shang, “New solitary wave solutions with compact support for the KdV-like K(m,n) equations with fully nonlinear dispersion,” Applied Mathematics and Computation, vol. 173, no. 2, pp. 1124-1136, 2006. · Zbl 1088.65094 |

[15] | A. Biswas, “1-soliton solution of the K(m,n) equation with generalized evolution and time-dependent damping and dispersion,” Computers & Mathematics with Applications, vol. 59, no. 8, pp. 2536-2540, 2010. · Zbl 1193.35181 |

[16] | A. Biswas, “1-soliton solution of the K(m,n) equation with generalized evolution,” Physics Letters. A, vol. 372, no. 25, pp. 4601-4602, 2008. · Zbl 1221.35099 |

[17] | H. Triki and A. M. Wazwaz, “Bright and dark soliton solutions for a K(m,n) equation with t-dependent coefficients,” Physics Letters A, vol. 373, no. 25, pp. 2162-2165, 2009. · Zbl 1229.35232 |

[18] | A. M. Wazwaz, “General solutions with solitary patterns for the defocusing branch of the nonlinear dispersive K(m,n) equations in higher dimensional spaces,” Applied Mathematics and Computation, vol. 133, no. 2-3, pp. 229-244, 2002. · Zbl 1027.35118 |

[19] | A. M. Wazwaz, “General compactons solutions for the focusing branch of the nonlinear dispersive K(n,n) equations in higher-dimensional spaces,” Applied Mathematics and Computation, vol. 133, no. 2-3, pp. 213-227, 2002. · Zbl 1027.35117 |

[20] | A. M. Wazwaz, “The tanh method for a reliable treatment of the K(n,n) and the KP(n,n) equations and its variants,” Applied Mathematics and Computation, vol. 170, no. 1, pp. 361-379, 2005. · Zbl 1082.65586 |

[21] | A. M. Wazwaz, “Compactons dispersive structures for variants of the K(n,n) and the KP equations,” Chaos, Solitons and Fractals, vol. 13, no. 5, pp. 1053-1062, 2002. · Zbl 0997.35083 |

[22] | A. M. Wazwaz, “The tanh-coth method for new compactons and solitons solutions for the K(n,n) and the K(n+1,n+1) equations,” Applied Mathematics and Computation, vol. 188, no. 2, pp. 1930-1940, 2007. · Zbl 1119.65101 |

[23] | Z. Feng, “Computations of soliton solutions and periodic solutions for the focusing branch of the nonlinear dispersive K(n,n) equations in higher-dimensional spaces,” Applied Mathematics and Computation, vol. 182, no. 1, pp. 781-790, 2006. · Zbl 1107.65088 |

[24] | T. A. Abassy, H. El Zoheiry, and M. A. El-Tawil, “A numerical study of adding an artificial dissipation term for solving the nonlinear dispersive equations K(n,n),” Journal of Computational and Applied Mathematics, vol. 232, no. 2, pp. 388-401, 2009. · Zbl 1173.65053 |

[25] | X. Deng, E. J. Parkes, and J. Cao, “Exact solitary and periodic-wave solutions of the K(2,2) equation (defocusing branch),” Applied Mathematics and Computation, vol. 217, no. 4, pp. 1566-1576, 2010. · Zbl 1202.35185 |

[26] | M. S. Ismail and T. R. Taha, “A numerical study of compactons,” Mathematics and Computers in Simulation, vol. 47, no. 6, pp. 519-530, 1998. · Zbl 0932.65096 |

[27] | M. S. Ismail, “A finite difference method for Korteweg-de Vries like equation with nonlinear dispersion,” International Journal of Computer Mathematics, vol. 74, no. 2, pp. 185-193, 2000. · Zbl 0955.65065 |

[28] | M. S. Ismail and F. R. Al-Solamy, “A numerical study of K(3,2) equation,” International Journal of Computer Mathematics, vol. 76, no. 4, pp. 549-560, 2001. · Zbl 0988.65078 |

[29] | J. D. Frutos, M. A. Lopez Marcos, and J. M. Sanz-Serna, “A finite difference scheme for the K(2,2) compacton equation,” Journal of Computational Physics, vol. 120, no. 2, pp. 248-252, 1995. · Zbl 0840.65090 |

[30] | J. B. Zhou and L. X. Tian, “Soliton solution of the osmosis K(2,2) equation,” Physics Letters. A, vol. 372, no. 41, pp. 6232-6234, 2008. · Zbl 1225.35194 |

[31] | C. H. Xu and L. X. Tian, “The bifurcation and peakon for K(2,2) equation with osmosis dispersion,” Chaos, Solitons and Fractals, vol. 40, no. 2, pp. 893-901, 2009. · Zbl 1197.35253 |

[32] | J. B. Zhou, L. X. Tian, and X. H. Fan, “New exact travelling wave solutions for the K(2,2) equation with osmosis dispersion,” Applied Mathematics and Computation, vol. 217, no. 4, pp. 1355-1366, 2010. · Zbl 1203.35258 |

[33] | B. He and Q. Meng, “New exact explicit peakon and smooth periodic wave solutions of the K(3,2) equation,” Applied Mathematics and Computation, vol. 217, no. 4, pp. 1697-1703, 2010. · Zbl 1203.35223 |

[34] | W. Rui, B. He, Y. Long, and C. Chen, “The integral bifurcation method and its application for solving a family of third-order dispersive PDEs,” Nonlinear Analysis. Theory, Methods & Applications, vol. 69, no. 4, pp. 1256-1267, 2008. · Zbl 1144.35461 |

[35] | W. Rui, S. Xie, B. He, and Y. Long, “Integral bifurcation method and its application for solving the modified equal width wave equation and its variants,” Rostocker Mathematisches Kolloquium, no. 62, pp. 87-106, 2007. · Zbl 1148.35079 |

[36] | W. Rui, L. Yao, Y. Long, B. He, and Z. Li, “Integral bifurcation method combined with computer for solving a higher order wave equation of KdV type,” International Journal of Computer Mathematics, vol. 87, no. 1-3, pp. 119-128, 2010. · Zbl 1182.65161 |

[37] | W. Rui and Y. Y. Lee, “Nonlinear single degree of freedom system with dynamic responses of smooth, non-smooth, and distorted travelling waves,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 9, pp. 1227-1235, 2009. · Zbl 06942496 |

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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.