Search
This Journal
This Journal
Anywhere
Quick Search in Journals
Enter words / phrases / DOI / ISBN / keywords / authors / etc
Access type:Only show content I have full access toOnly show Open Access
Advanced Search
0 My Cart
Sign in
Institutional Access

System Upgrade on Tue, May 28th, 2024 at 2am (EDT)

Existing users will be able to log into the site and access content. However, E-commerce and registration of new users may not be available for up to 12 hours.
For online purchase, please visit us again. Contact us at customercare@wspc.com for any enquiries.

PapersNo Access

COEXISTENCE OF MULTIFARIOUS EXPLICIT NONLINEAR WAVE SOLUTIONS FOR MODIFIED FORMS OF CAMASSA–HOLM AND DEGAPERIS–PROCESI EQUATIONS

Department of Mathematics, South China University of Technology, Guangzhou, Guangdong 510640, P. R. China

Author for correspondence.

Search for more papers by this author

and

School of Civil Engineering and Transportation, South China University of Technology, Guangzhou, Guangdong 510640, P. R. China

Search for more papers by this author

https://doi.org/10.1142/S0218127409024050Cited by:11 (Source: Crossref)

Abstract

In this paper, we consider two nonlinear equations u_t - u_xxt + 3u²u_x = 2u_xu_xx + uu_xxx and u_t - u_xxt + 4u²u_x = 3u_xu_xx + uu_xxx which are called modified forms of Camassa–Holm and Degaperis–Procesi equations, respectively. For given constant wave speed, we investigate the coexistence of multifarious explicit nonlinear wave solutions, solitary wave solution, peakon wave solution, singular wave solution, smooth periodic wave solution and singular periodic wave solution. Not only is the coexistence shown, but the concrete expressions are presented via the bifurcation method of dynamical systems. From our work it can be seen that these two equations possess a lot of similar properties, and the types of their explicit nonlinear wave solutions are more than that of Camassa–Holm and Degaperis–Procesi equations. Many previous results are our special cases.

Research is supported by the National Natural Science Foundation of China (No. 10871073) and Guangdong Province (No. 07006552).

Keywords:

References

S. Abbasbandy and E. J. Parkes, Chaos Solit. Fract. 36, 581 (2008), DOI: 10.1016/j.chaos.2007.10.034. Crossref, Web of Science, Google Scholar
A. Bressan and A. Constantin, Arch. Rat. Mech. Anal. 183, 215 (2007), DOI: 10.1007/s00205-006-0010-z. Crossref, Web of Science, Google Scholar
R. Camassa and D. D. Holm, Phys. Rev. Lett. 71, 1661 (1993), DOI: 10.1103/PhysRevLett.71.1661. Crossref, Web of Science, Google Scholar
C. Chen and M. Y. Tang, Chaos Solit. Fract. 27, 698 (2006), DOI: 10.1016/j.chaos.2005.04.040. Crossref, Web of Science, Google Scholar
G. M. Coclite and K. H. Karlsen, J. Diff. Eqs. 234, 142 (2007), DOI: 10.1016/j.jde.2006.11.008. Crossref, Web of Science, Google Scholar
G. M. Coclite, K. H. Karlsen and N. H. Risebro, IMA J. Numer. Anal. 28, 80 (2008), DOI: 10.1093/imanum/drm003. Crossref, Web of Science, Google Scholar
A. Constantin and W. A. Strauss, Commun. Pure Appl. Math. 53, 603 (2000). Crossref, Web of Science, Google Scholar
A. Constantin and L. Molinet, Physica D 157, 75 (2001), DOI: 10.1016/S0167-2789(01)00298-6. Crossref, Web of Science, Google Scholar
A. Constantin, V. S. Gerdjikov and R. I. Ivanov, Inver. Probl. 22, 2197 (2006), DOI: 10.1088/0266-5611/22/6/017. Crossref, Web of Science, Google Scholar
A. Constantin, Philos. Trans. Roy. Society A-Math. Phys. Engin. Sci. 365, 2195 (2007), DOI: 10.1098/rsta.2007.2001. Crossref, Web of Science, Google Scholar
A. Degasperis and M. Procesi, Symmetry and Perturbation Theory, eds. A. Degasperis and G. Gaeta (World Scientific, Singapore, 1999) pp. 23–37. Google Scholar
J. Escher, Y. Liu and Z. Y. Yin, Indiana Univ. Math. J. 56, 87 (2007), DOI: 10.1512/iumj.2007.56.3040. Crossref, Web of Science, Google Scholar
J. Escher and Z. Y. Yin, Commun. Partial Diff. Eqs. 33, 377 (2008), DOI: 10.1080/03605300701318872. Crossref, Web of Science, Google Scholar
F. Gesztesy and H. Holden, Philos. Trans. Roy. Soc. A-Math. Phys. Engin. Sci. 366, 1025 (2008), DOI: 10.1098/rsta.2007.2060. Crossref, Web of Science, Google Scholar
J. S. Kamdem and Z. J. Qiao, Chaos Solit. Fract. 31, 437 (2007), DOI: 10.1016/j.chaos.2005.09.071. Crossref, Web of Science, Google Scholar
D. J. Kaup, Studies Appl. Math. 117, 149 (2006), DOI: 10.1111/j.1467-9590.2006.00350.x. Crossref, Web of Science, Google Scholar
S. A. Khuri, Chaos Solit. Fract. 25, 705 (2005), DOI: 10.1016/j.chaos.2004.11.083. Crossref, Web of Science, Google Scholar
J. Lenells, J. Math. Anal. Appl. 306, 72 (2005), DOI: 10.1016/j.jmaa.2004.11.038. Crossref, Web of Science, Google Scholar
C. S. Liu, Chinese Phys. 16, 1832 (2007). Google Scholar
Z. R. Liu and T. F. Qian, Int. J. Bifurcation and Chaos 11, 781 (2001). Link, Web of Science, Google Scholar
Z. R. Liu, A. M. Kayed and C. Chen, Int. J. Bifurcation and Chaos 16, 2261 (2006). Link, Web of Science, Google Scholar
Z. R. Liu and Y. Long, Nonlin. Anal.: Real World Appl. 8, 136 (2007), DOI: 10.1016/j.nonrwa.2005.06.005. Crossref, Web of Science, Google Scholar
Z. R. Liu and Z. Y. Ouyang, Phys. Lett. A 366, 377 (2007), DOI: 10.1016/j.physleta.2007.01.074. Crossref, Web of Science, Google Scholar
Z. R. Liu and B. L. Guo, Progr. Natural Sci. 18, 259 (2008), DOI: 10.1016/j.pnsc.2007.11.004. Crossref, Web of Science, Google Scholar
Lundmark, H. & Szmigielski, J. [2005] "Degasperis–Procesi peakons and the discrete cubic string," Int. Math. Res. Papers (2), 53–116 . Google Scholar
H. Lundmark, J. Nonl. Sci. 17, 169 (2007), DOI: 10.1007/s00332-006-0803-3. Crossref, Web of Science, Google Scholar
T. R. Marchant and N. F. Smyth, Phys. Rev. E 73, 057602 (2006), DOI: 10.1103/PhysRevE.73.057602. Crossref, Web of Science, Google Scholar
Y. Matsuno, Inve. Probl. 21, 1553 (2005), DOI: 10.1088/0266-5611/21/5/004. Crossref, Web of Science, Google Scholar
Y. Matsuno, Inver. Probl. 21, 2085 (2005), DOI: 10.1088/0266-5611/21/6/018. Crossref, Web of Science, Google Scholar
Y. Matsuno, Phys. Lett. A 359, 451 (2006), DOI: 10.1016/j.physleta.2006.06.065. Crossref, Web of Science, Google Scholar
A. Parker and Y. Matsuno, J. Phys. Soc. Japan 75, 124001 (2006). Crossref, Google Scholar
J. W. Shen and W. Xu, Chaos Solit. Fract. 26, 1149 (2005), DOI: 10.1016/j.chaos.2005.02.021. Crossref, Web of Science, Google Scholar
L. X. Tian and X. Y. Song, Chaos Solit. Fract. 21, 621 (2004). Google Scholar
Q. D. Wang and M. Y. Tang, Phys. Lett. A 372, 2995 (2008), DOI: 10.1016/j.physleta.2008.01.012. Crossref, Web of Science, Google Scholar
A. M. Wazwaz, Phys. Lett. A 352, 500 (2006), DOI: 10.1016/j.physleta.2005.12.036. Crossref, Web of Science, Google Scholar
A. M. Wazwaz, Appl. Math. Comput. 186, 130 (2007), DOI: 10.1016/j.amc.2006.07.092. Crossref, Web of Science, Google Scholar
E. Yomba, Phys. Lett. A 372, 215 (2008), DOI: 10.1016/j.physleta.2007.03.008. Crossref, Web of Science, Google Scholar
E. Yomba, Phys. Lett. A 372, 1048 (2008), DOI: 10.1016/j.physleta.2007.09.003. Crossref, Web of Science, Google Scholar

Journal cover image

Vol. 19, No. 07

Metrics

Downloaded 28 times

History

Received 22 July 2008

Revised 22 September 2008

Keywords