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Smooth and non-smooth traveling waves in a nonlinearly dispersive equation. (English) Zbl 0985.37072
Summary: 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.

MSC:
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
35Q58Other completely integrable PDE (MSC2000)
35B65Smoothness and regularity of solutions of PDE
76B25Solitary waves (inviscid fluids)
34C37Homoclinic and heteroclinic solutions of ODE