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Zbl 1232.35125
Fan, Xinghua; Yang, Shouxiang; Yin, Jiuli; Tian, Lixin
Bifurcations of traveling wave solutions for a two-component Fornberg-Whitham equation. (English)
[J] Commun. Nonlinear Sci. Numer. Simul. 16, No. 10, 3956-3963 (2011). ISSN 1007-5704

The authors study a two-component Fornberg-Whitham equation given by $u_t=u_{xxt}-u_x-uu_x+3u_xu_{xx}+uu_{xxx}+\rho_x$, $\rho_t=-(\rho u)_x$, where $u=u(x,t)$ is the height of the water surface above a horizontal bottom, and $\rho=\rho(x,t)$ is related to the horizontal velocity field. Under additional conditions it is shown that there are smooth solutions, non smooth solutions and periodic wave solutions. The proofs are based on transforming the Fornberg-Whitham system into a planar dynamical system and on a discussion of phase portraits. Moreover, the authors present some explicit solutions.
[Erich Miersemann (Leipzig)]

MSC 2000:: *35Q35 Other equations arising in fluid mechanics
35Q53 KdV-like equations
35C07
76B15 Wave motions (fluid mechanics)
35B65 Smoothness of solutions of PDE

Keywords: two-component Fornberg-Whitham equation; solitary wave solutions; bifurcation method; soliton; kink solution; antikink solution

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