##
**Trigonometric function periodic wave solutions and their limit forms for the KdV-like and the PC-like equations.**
*(English)*
Zbl 1235.35259

Summary: We use the bifurcation method of dynamical systems to study the periodic wave solutions and their limit forms for the KdV-like equation \(u_t + a(1 + bu)uu_x + u_{xxx} = 0\), and PC-like equation \(v_{tt} - v_{ttxx} - (a_1 v + a_2 v^2 + a_3 v^3)_{xx} = 0\), respectively. Via some special phase orbits, we obtain some new explicit periodic wave solutions which are called trigonometric function periodic wave solutions because they are expressed in terms of trigonometric functions. We also show that the trigonometric function periodic wave solutions can be obtained from the limits of elliptic function periodic wave solutions. It is very interesting that the two equations have similar periodic wave solutions.

### MSC:

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

35B10 | Periodic solutions to PDEs |

35B32 | Bifurcations in context of PDEs |

PDF
BibTeX
XML
Cite

\textit{L. Zhengrong} et al., Math. Probl. Eng. 2011, Article ID 810217, 23 p. (2011; Zbl 1235.35259)

Full Text:
DOI

### References:

[1] | B. Dey, “Domain wall solutions of KdV-like equations with higher order nonlinearity,” Journal of Physics A, vol. 19, no. 1, pp. L9-L12, 1986. · Zbl 0624.35070 |

[2] | B. Dey, “KdV like equations with domain wall solutions and their Hamiltonians,” in Solitons, pp. 188-194, Springer, Berlin, Germany, 1988. · Zbl 0694.35191 |

[3] | J. F. Zhang, “New solitary wave solution of the combined KdV and mKdV equation,” International Journal of Theoretical Physics, vol. 37, no. 5, pp. 1541-1546, 1998. · Zbl 0941.35098 |

[4] | J. F. Zhang, F. M. Wu, and J. Q. Shi, “Simple solition solution method for the combined KdV and mKdV equation,” International Journal of Theoretical Physics, vol. 39, no. 6, pp. 1697-1702, 2000. · Zbl 0941.35098 |

[5] | J. Yu, “Exact solitary wave solutions to a combined KdV and mKdV equation,” Mathematical Methods in the Applied Sciences, vol. 23, no. 18, pp. 1667-1670, 2000. · Zbl 0988.76014 |

[6] | R. Grimshaw, D. Pelinovsky, E. Pelinovsky, and A. Slunyaev, “Generation of large-amplitude solitons in the extended Korteweg-de Vries equation,” Chaos, vol. 12, no. 4, pp. 1070-1076, 2002. · Zbl 1080.35532 |

[7] | E. G. Fan, “Multiple travelling wave solutions of nonlinear evolution equations using a unified algebraic method,” Journal of Physics A, vol. 35, no. 32, pp. 6853-6872, 2002. · Zbl 1039.35029 |

[8] | E. G. Fan, “Uniformly constructing a series of explicit exact solutions to nonlinear equations in mathematical physics,” Chaos, Solitons and Fractals, vol. 16, no. 5, pp. 819-839, 2003. · Zbl 1030.35136 |

[9] | M. Y. Tang, R. Q. Wang, and Z. J. Jing, “Solitary waves and their bifurcations of KdV like equation with higher order nonlinearity,” Science in China A, vol. 45, no. 10, pp. 1255-1267, 2002. · Zbl 1099.37057 |

[10] | Y.-Z. Peng, “New exact solutions to the combined KdV and mKdV equation,” International Journal of Theoretical Physics, vol. 42, no. 4, pp. 863-868, 2003. · Zbl 1027.35110 |

[11] | K. W. Chow, R. H. J. Grimshaw, and E. Ding, “Interactions of breathers and solitons in the extended Korteweg-de Vries equation,” Wave Motion, vol. 43, no. 2, pp. 158-166, 2005. · Zbl 1231.35196 |

[12] | D. Kaya and I. E. Inan, “A numerical application of the decomposition method for the combined KdV-MKdV equation,” Applied Mathematics and Computation, vol. 168, no. 2, pp. 915-926, 2005. · Zbl 1080.65100 |

[13] | E. Yomba, “The extended Fan’s sub-equation method and its application to KdV-MKdV, BKK and variant Boussinesq equations,” Physics Letters A, vol. 336, no. 6, pp. 463-476, 2005. · Zbl 1136.35451 |

[14] | W. G. Zhang and W. X. Ma, “Explicit solitary-wave solutions to generalized Pochhammer-Chree equations,” Applied Mathematics and Mechanics, vol. 20, no. 6, pp. 625-632, 1999. · Zbl 0935.35132 |

[15] | J. B. Li and L. J. Zhang, “Bifurcations of traveling wave solutions in generalized Pochhammer-Chree equation,” Chaos, Solitons and Fractals, vol. 14, no. 4, pp. 581-593, 2002. · Zbl 0997.35096 |

[16] | D. Kaya, “The exact and numerical solitary-wave solutions for generalized modified Boussinesq equation,” Physics Letters A, vol. 348, no. 3-6, pp. 244-250, 2006. · Zbl 1195.35264 |

[17] | M. Rafei, D. D. Ganji, H. R. M. Daniali, and H. Pashaei, “Application of homotopy perturbation method to the RLW and generalized modified Boussinesq equations,” Physics Letters A, vol. 364, no. 1, pp. 1-6, 2007. · Zbl 1203.65214 |

[18] | R. Liu, “Some new results on explicit traveling wave solutions of k(m,n) equation,” Discrete and Continuous Dynamical Systems B, vol. 13, no. 3, pp. 633-646, 2010. · Zbl 1195.35092 |

[19] | M. Li, S. C. Lim, and S. Y. Chen, “Exact solution of impulse response to a class of fractional oscillators and its stability,” Mathematical Problems in Engineering, vol. 2011, Article ID 657839, 9 pages, 2011. · Zbl 1202.34018 |

[20] | J.-B. Li, “Exact traveling wave solutions and dynamical behavior for the (n+1)-dimensional multiple sine-Gordon equation,” Science in China A, vol. 50, no. 2, pp. 153-164, 2007. · Zbl 1214.35005 |

[21] | C. Cattani and G. Pierro, “Complexity on acute myeloid leukemia mRNA transcript variant,” Mathematical Problems in Engineering, vol. 2011, Article ID 379873, 16 pages, 2011. · Zbl 1235.92025 |

[22] | C. Cattani and A. Kudreyko, “Application of periodized harmonic wavelets towards solution of eigenvalue problems for integral equations,” Mathematical Problems in Engineering, vol. 2010, Article ID 570136, 8 pages, 2010. · Zbl 1191.65175 |

[23] | Z. R. Liu and Z. Y. Ouyang, “A note on solitary waves for modified forms of Camassa-Holm and Degasperis-Procesi equations,” Physics Letters A, vol. 366, no. 4-5, pp. 377-381, 2007. · Zbl 1203.35234 |

[24] | E. G. Bakhoum and C. Toma, “Specific mathematical aspects of dynamics generated by coherence functions,” Mathematical Problems in Engineering, vol. 2011, Article ID 436198, 10 pages, 2011. · Zbl 1248.37075 |

[25] | M.-Y. Tang and W.-L. Zhang, “Four types of bounded wave solutions of CH-\gamma equation,” Science in China A, vol. 50, no. 1, pp. 132-152, 2007. · Zbl 1117.35310 |

[26] | Q.-D. Wang and M.-Y. Tang, “New explicit periodic wave solutions and their limits for modified form of Camassa-Holm equation,” Acta Mathematicae Applicatae Sinica, vol. 26, no. 3, pp. 513-524, 2010. · Zbl 1223.34070 |

[27] | E. G. Bakhoum and C. Toma, “Mathematical transform of traveling-wave equations and phase aspects of quantum interaction,” Mathematical Problems in Engineering, vol. 2010, Article ID 695208, 15 pages, 2010. · Zbl 1191.35220 |

[28] | R. Liu, “Several new types of solitary wave solutions for the generalized Camassa-Holm-Degasperis-Procesi equation,” Communications on Pure and Applied Analysis, vol. 9, no. 1, pp. 77-90, 2010. · Zbl 1196.34002 |

[29] | M. Li, “Fractal time series-a tutorial review,” Mathematical Problems in Engineering, vol. 2010, Article ID 157264, 26 pages, 2010. · Zbl 1191.37002 |

[30] | S. Y. Chen, H. Tong, Z. Wang, S. Liu, M. Li, and B. Zhang, “Improved generalized belief propagation for vision processing,” Mathematical Problems in Engineering, vol. 2011, Article ID 416963, 12 pages, 2011. · Zbl 1202.94026 |

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.