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Bifurcations of smooth and non-smooth travelling wave solutions in the generalized Camassa-Holm equation. (English) Zbl 1072.35579
Summary: The dynamical behavior of travelling wave solutions to the generalized Camassa-Holm equation u t +2ku x -u xxt +au m u x =2u x u xx +uu xxx is analyzed by using bifurcation theory and the method of phase portraits analysis. The condition under which compactons and cusp waves appear are also given. In addition, the reason for solitary cusp wave and periodic cusp wave to exist is highlighted.
MSC:
35Q53KdV-like (Korteweg-de Vries) equations
34C23Bifurcation (ODE)
35B32Bifurcation (PDE)
37K50Bifurcation problems (infinite-dimensional systems)