Li, Jibin; Li, Yishen Bifurcations of travelling wave solutions for a two-component Camassa-Holm equation. (English) Zbl 1182.34064 Acta Math. Sin., Engl. Ser. 24, No. 8, 1319-1330 (2008). The following two-component generalization of the Camassa-Holm equation is considered\[ m_t + u m_x + 2mu_x+ e\rho\rho_x=0, \]\[ \rho_t + (\rho u)_x=0, \]where \(e=\pm 1\), \(m=u -\alpha^2 u_{xx}-k\), for \(\alpha, k\) real parameters.The existence of solitary wave solutions (homoclinic orbits), kink and anti-kink wave solutions (heteroclinic orbits) and periodic wave solutions is investigated by using a dynamical systems approach. Some exact explicit parametric representations of travelling wave solutions are provided too.Finally, the existence of uncountably infinite many breaking wave solutions (whose maximal existence interval is bounded) is proved for \(e=-1\). Reviewer: Cristina Marcelli (Ancona) Cited in 18 Documents MSC: 34C37 Homoclinic and heteroclinic solutions to ordinary differential equations 34C23 Bifurcation theory for ordinary differential equations 35Q53 KdV equations (Korteweg-de Vries equations) 35Q51 Soliton equations 37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems 37L05 General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations Keywords:solitary waves; kink wave solutions; periodic wave solutions; breaking wave solutions; smoothness of wave PDFBibTeX XMLCite \textit{J. Li} and \textit{Y. Li}, Acta Math. Sin., Engl. Ser. 24, No. 8, 1319--1330 (2008; Zbl 1182.34064) Full Text: DOI References: [1] Chen, M., Liu, S. Q., Zhang, Y. J.: A 2-component generalization of the Cammassa-Holm equation and its solution. Letters in Math. Phys., 75, 1–15 (2006) · Zbl 1105.35102 [2] Cammasa, R., Holm, D. D.: An integrable shallow water equation with peaked solution. Phys. Rev. Lett., 71, 1161–1164 (1993) [3] Cammasa, R., Holm, D. D., Hyman, J. M.: A new integrable shallow water equation. Adv. Appl. Mech., 31, 1–33 (1994) · Zbl 0808.76011 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.