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:
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 35Q58 | Other completely integrable PDE (MSC2000) |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 76B25 | Solitary waves for incompressible inviscid fluids |
| 34C37 | Homoclinic and heteroclinic solutions to ordinary differential equations |
Keywords:
solitary waves;periodic waves;integrable system;bifurcations of phase portraits;smoothness of wavesCite
Full Text:DOI
References:
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