Abstract
The method of the phase plane is employed to investigate the solitary and periodic traveling waves in a nonlinear dispersive integrable partial differential equation. It is shown that the existence of a singular straight line in the corresponding ordinary differential equation for traveling wave solutions is the reason that smooth solitary wave solutions converge to solitary cusp wave solutions when the parameters are varied. The different parameter conditions for the existence of different kinds of solitary and periodic wave solutions are rigorously determined.
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