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Bifurcations of traveling wave solutions for a two-component Fornberg-Whitham equation. (English) Zbl 1232.35125
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.

35Q35PDEs in connection with fluid mechanics
35Q53KdV-like (Korteweg-de Vries) equations
35C07Traveling wave solutions of PDE
76B15Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35B65Smoothness and regularity of solutions of PDE
Full Text: DOI
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