Single peak solitary wave and compacton solutions of the generalized two-component Hunter–Saxton system
Dynamical system theory is applied to the generalized two-component Hunter–Saxton system. Two singular straight lines are found in the associated topological vector field. The influence of parameters as well as the singular lines on the smoothness property of the traveling wave solutions is explored in detail. We obtain the single peak solitary wave and compacton solutions for the generalized two-component Hunter–Saxton system. Asymptotic analysis and numerical simulations are provided for smooth solitary wave, peakon, cuspon and compacton solutions of the generalized two-component Hunter–Saxton system.
KeywordsHunter–Saxton system Solitary wave Peakon Cuspon Compacton
This work are supported by by the National Natural Science Foundation of China (No. 11161013 and No. 11361017), Guangxi Natural Science Foundation (No. 2014GXNSFBA118007), Foundation of Guangxi Key Lab of Trusted Software and Program for Innovative Research Team of Guilin University of Electronic Technology. The authors wish to thank the anonymous reviewers for their helpful comments and suggestions.
- 8.Li, J.B., Qiao, Z.J.: Bifurcations and exact traveling wave solutions of the generialized two-component Camassa-Holm equation, Inter. J. Bifurc. Chaos 22, 1250505-1-13 (2012)Google Scholar
- 16.Zhang, J., Tian, L.X.: Wave-breaking criterion for the generalized weakly dissipative periodic two-component Hunter-Saxton system. J. Appl. Math. 2013, 809824-1-10 (2013)Google Scholar
- 18.Li, J.B., Qiao, Z.J.: Peakon, pseudo-peakon, and cuspon solutions for two generalized Camassa-Holm equations, J. Math. Phys. 54, 123501-1-14 (2013)Google Scholar
- 21.Liu, H., Fang, Y., Xu, C.: The bifurcation and exact travelling wave solutions of (1+2)-dimensional nonlinear Schrodinger equation with dual-power law nonlinearity. Nonlinear Dyn. 67, 465–473 (2012)Google Scholar
- 22.Wang, Y., Bi, Q.: Different wave solutions associated with singular lines on phase plane. Nonlinear Dyn. 69, 1705–1731 (2012)Google Scholar
- 24.Chen, A.Y., et al.: Effects of quadratic singular curves in integrable equations. Stud. Appl. Math. DOI: 10.1111/sapm.12060
- 25.Chen, A.Y., Zhu, W.J., Qiao, Z.J., Huang, W.T.: Algebraic traveling wave solutions of a non-local hydrodynamic-type model. Math. Phys. Anal. Geom. to appearGoogle Scholar