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Volume 8, Number 6, 2018, Pages 1938-1958                                                                DOI:10.11948/2018.1938
Peakon and cuspon solutions of a generalized Camassa-Holm-Novikov equation
Lijun Zhang,Yue Wang,Chaudry Masood Khalique,Yuzhen Bai
Keywords:Generalized Camassa-Holm-Novikov equation, dynamical system approach, bifurcation, singular wave solutions.
      The bounded traveling wave solutions of a generalized Camassa-Holm-Novikov equation with $p=2$ and $p=3$ are derived via the dynamical system approach. The singular wave solutions including peakons and cuspons are obtained by the bifurcation analysis of the corresponding singular dynamical system and the orbits intersecting with or approaching the singular lines. The results show that the generalized Camassa-Holm-Novikov equation with $p=2$ and $p=3$ both admit smooth solitary wave, smooth periodic wave solutions, solitary peakons, periodic peakons, solitary cuspons and periodic cuspons as well. It is worth pointing out that the Novikov equation has no bounded traveling wave solutions with negative wave speed, but has a family of new periodic cuspons which are distinguished with the normal periodic cuspons for their discontinuous first-order derivatives at both maximum and minimum.
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