×

Bifurcations of smooth and nonsmooth traveling wave solutions in a generalized Degasperis-Procesi equation. (English) Zbl 1129.34032

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.
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.

MSC:

34C23 Bifurcation theory for ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
35B65 Smoothness and regularity of solutions to PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Camassa, R.; Holm, D. D., An integrable shallow water equation with peaked solitons [J], Phys. Rev. Lett., 71, 11, 1661-1664 (1993) · Zbl 0972.35521
[2] Chow, S. N.; Hale, J. K., Method of Bifurcation Theory (1981), Springer: Springer New York
[3] Cooper, F.; Shepard, H., Solitons in the Camassa-Holm shallow water equation, Phys. Lett. A, 194, 4, 246-250 (1994) · Zbl 0961.76512
[4] A. Degasperis, M. Procesi, in: A. Degasperis, G. Gaeta, (Eds.), Asymptotic Integrability Symmetry and Perturbation Theory, World Scientific, Singapore, 1999, pp. 23-37.; A. Degasperis, M. Procesi, in: A. Degasperis, G. Gaeta, (Eds.), Asymptotic Integrability Symmetry and Perturbation Theory, World Scientific, Singapore, 1999, pp. 23-37. · Zbl 0963.35167
[5] Degasperis, A., A new integrable equation with peakon solutions, Theoret. Math. Phys., 133, 1461-1472 (2002)
[6] Guckenheimer, J.; Holmes, P. J., Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (1983), Springer: Springer New York · Zbl 0515.34001
[7] Guo, B. L.; Liu, Z. R., Periodic cusp wave solutions and single-solutions for the b-equation, Chaos Solitons Fractals, 23, 1451-1463 (2005) · Zbl 1068.35103
[8] Lenells, J., Traveling wave solutions of the Degasperis-Procesi equation, J. Math. Anal. Appl., 306, 72-82 (2005) · Zbl 1068.35163
[9] Li, J.; Liu, X., Smooth and non-smooth traveling waves in a nonlinearly dispersion equation, Appl. Math. Model, 25, 41-56 (2000) · Zbl 0985.37072
[10] Perko, L., Differential Equations and Dynamical Systems (1991), Springer: Springer New York · Zbl 0717.34001
[11] Shen, J.; Xu, W., Bifurcations of smooth and non-smooth traveling wave solutions in the generalized Camassa-Holm equation, Chaos Solitons Fractals, 26, 1149-1162 (2005) · Zbl 1072.35579
[12] Vakhnenko, V. O.; Parkes, E. J., Periodic and solitary-wave solutions of the Degasperis-Procesi equation, Chaos Solitons Fractals, 20, 1059-1073 (2004) · Zbl 1049.35162
[13] L. Zhang, L.-Q. Chen, X. Huo, Peakons and periodic cusp wave solutions in a generalized Camassa-Holm equation, Chaos Solitons Fractals (2006), to appear.; L. Zhang, L.-Q. Chen, X. Huo, Peakons and periodic cusp wave solutions in a generalized Camassa-Holm equation, Chaos Solitons Fractals (2006), to appear. · Zbl 1142.35591
[14] Zhang, L.; Li, J., Bifurcations of traveling wave solutions in a coupled non-linear wave equation, Chaos Solitons Fractals, 17, 941-950 (2003) · Zbl 1030.35142
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.