Song, Ming; Cao, Jun; Guan, Xiaoli Application of the bifurcation method to the Whitham-Broer-Kaup-like equations. (English) Zbl 1255.35182 Math. Comput. Modelling 55, No. 3-4, 688-696 (2012). Summary: We investigate the Whitham-Broer-Kaup-like equations. Some explicit expressions of solutions for the equation are obtained by using the bifurcation method and qualitative theory of dynamical systems. These solutions contain kink-shaped solutions, blow-up solutions, periodic blow-up solutions and solitary wave solutions. Some previous results are extended. Cited in 12 Documents MSC: 35Q35 PDEs in connection with fluid mechanics 35B32 Bifurcations in context of PDEs 35B44 Blow-up in context of PDEs 35B10 Periodic solutions to PDEs 35C08 Soliton solutions Keywords:bifurcation method; Whitham-Broer-Kaup-like equations; exact solutions PDFBibTeX XMLCite \textit{M. Song} et al., Math. Comput. Modelling 55, No. 3--4, 688--696 (2012; Zbl 1255.35182) Full Text: DOI References: [1] ZHOU, Y. B.; Li, C., Application of modified \(G^\prime / G\)-expansion method to traveling wave solutions for Whitham-Broer-Kaup-Like equations, Commun. Theoret. Phys. (Beijing), 51, 664-670 (2009) · Zbl 1181.35223 [2] Guo, S. M.; Zhou, Y. B., The extended \(\frac{G^\prime}{G} \)-expansion method and its applications to the Whitham-Broer-Kaup-Like equations and coupled Hirota-Satsuma KdV equations, Appl. Math. Comput., 215, 3214-3221 (2010) · Zbl 1187.35209 [3] Xie, F. D.; Yan, Z. Y.; Zhang, H. Q., Explicit and exact traveling wave solutions of Whitham-Broer-Kaup shallow water equations, Phys. Lett. A, 285, 76-80 (2001) · Zbl 0969.76517 [4] Chen, Y.; Wang, Q.; Li, B., Elliptic equation rational expansion method and new exact travelling solutions for Whitham-Broer-Kaup equations, Chaos Solitons Fractals, 26, 231-246 (2005) · Zbl 1080.35080 [5] El-Sayed, S. M.; Kaya, D., Exact and numerical traveling wave solutions of Whitham-Broer-Kaup equations, Appl. Math. Comput., 167, 1339-1349 (2005) · Zbl 1082.65580 [6] Yomba, E., The extended Fans sub-equation method and its application to KdV-MKdV, BKK and variant Boussinesq equations, Phys. Lett. A, 336, 463-476 (2005) · Zbl 1136.35451 [7] Yan, Z. Y.; Zhang, H. Q., New explicit and exact travelling wave solutions for a system of variant Boussinesq equations in mathematical physics, Phys. Lett. A, 252, 291-296 (1999) · Zbl 0938.35130 [8] Chen, Y.; Wang, Q., A new general algebraic method with symbolic computation to construct new travelling wave solution for the (1+1)-dimensional dispersive long wave equation, Appl. Math. Comput., 168, 1189-1204 (2005) · Zbl 1082.65578 [9] Zheng, X. D.; Chen, Y.; Zhang, H. Q., Generalized extended tanh-function method and its application to (1+1)-dimensional dispersive long wave equation, Phys. Lett. A, 311, 145-157 (2003) · Zbl 1019.35059 [10] Li, J. B.; Liu, Z. R., Smooth and non-smooth traveling waves in a nonlinearly dispersive equation, Appl. Math. Model., 25, 41-56 (2000) · Zbl 0985.37072 [11] Liu, Z. R.; Qian, T. F., Peakons and their bifurcation in a generalized Camassa-Holm equation, Internat. J. Bifur. Chaos, 11, 781-792 (2001) · Zbl 1090.37554 [12] Liu, Z. R.; Yang, C. X., The application of bifurcation method to a higher-order KDV equation, J. Math. Anal. Appl., 275, 1-12 (2002) · Zbl 1012.35076 [13] Tang, M. Y.; Wang, R. Q.; Jing, Z. J., Solitary waves and their bifurcations of KdV like equation with higher order nonlinearity, Sci. China Ser. A, 45, 1255-1267 (2002) · Zbl 1099.37057 [14] Liu, Z. R.; Guo, B. L., Periodic blow-up solutions and their limit forms for the generalized Camassa-Holm equation, Prog. Nat. Sci., 18, 259-266 (2008) [15] Song, M.; Yang, C. X.; Zhang, B. G., Exact solitary wave solutions of the Kadomtsov-Petviashvili-Benjamin-Bona-Mahony equation, Appl. Math. Comput., 217, 1334-1339 (2010) · Zbl 1203.35245 [16] Song, M.; Cai, J. H., Solitary wave solutions and kink wave solutions for a generalized Zakharov-Kuznetsov equation, Appl. Math. Comput., 217, 1455-1462 (2010) · Zbl 1203.35244 [17] Wen, Z. S.; Liu, Z. R.; Song, M., New exact solutions for the classical Drinfel’d-Sokolov-Wilson equation, Appl. Math. Comput., 215, 2349-2358 (2009) · Zbl 1181.35221 [18] Song, M.; Yang, C. X., Exact traveling wave solutions of the Zakharov-Kuznetsov-Benjamin-Bona-Mahony equation, Appl. Math. Comput., 216, 3234-3243 (2010) · Zbl 1193.35199 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.