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A type of bounded traveling wave solutions for the Fornberg-Whitham equation. (English) Zbl 1146.35025

Summary: By using bifurcation method, we successfully find the Fornberg-Whitham equation

${u}_{t}-{u}_{xxt}+{u}_{x}=u{u}_{xxx}-u{u}_{x}+3{u}_{x}{u}_{xx}$

has a type of traveling wave solutions called kink-like wave solutions and antikink-like wave solutions. They are defined on some semifinal bounded domains and possess properties of kink waves and anti-kink waves. Their implicit expressions are obtained. For some concrete data, the graphs of the implicit functions are displayed, and the numerical simulation is made. The results show that our theoretical analysis agrees with the numerical simulation.

##### MSC:
 35G25 Initial value problems for nonlinear higher-order PDE