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Zbl 1129.34032
Zhang, Lijun; Chen, Li-Qun; Huo, Xuwen
Bifurcations of smooth and nonsmooth traveling wave solutions in a generalized Degasperis-Procesi equation.
(English)
[J] J. Comput. Appl. Math. 205, No. 1, 174-185 (2007). ISSN 0377-0427

The authors employ bifurcation theory of planar dynamical systems to study smooth and nonsmooth travelling wave solutions of the generalized Degasperis-Procesi equation $$u_t - u_{xxt} + 4 u^m u_x = 3u_xu_{xx} + uu_{xxx}.$$ By the usual ansatz $u=\phi(x-ct)$ and integrating the resulting ODE once, they obtain a singular second order differential equation depending on 3 parameters $m$, $c$ and an integration constant $g$. The periodic, homoclinic and heteroclinic orbits in the reduced system correspond to travelling waves in the original PDE. Due to the singularity of the ODE these travelling waves may lose smoothness. \par The different possibilities of phase portraits for even and odd values of $m$ are classified for the different values of the parameters $c$ and $g$. Numerical calculations show good agreement with the analytical results.
[Alois Steindl (Wien)]
MSC 2000:
*34C23 Bifurcation (periodic solutions)
34C25 Periodic solutions of ODE
34C37 Homoclinic and heteroclinic solutions of ODE
35B65 Smoothness of solutions of PDE
35Q53 KdV-like equations

Keywords: GDP equation; compacton; peakon; periodic cusp wave; phase portrait

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