Hostname: page-component-cb9f654ff-hqlzj Total loading time: 0 Render date: 2025-08-23T14:18:04.862Z Has data issue: false hasContentIssue false
  • English
  • Français

Bifurcations and Single Peak Solitary Wave Solutions of an Integrable Nonlinear Wave Equation

Part of: Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  19 September 2016

Wei Wang ,
Chunhai Li  and
Wenjing Zhu
Wei Wang
Affiliation:
School of Mathematics and Computing Science, Guilin University of Electronic Technology, Jinji Road No. 1, Guilin 541004, China
Chunhai Li*
Affiliation:
School of Mathematics and Computing Science, Guilin University of Electronic Technology, Jinji Road No. 1, Guilin 541004, China
Wenjing Zhu
Affiliation:
School of Mathematics and Computing Science, Guilin University of Electronic Technology, Jinji Road No. 1, Guilin 541004, China
*
*Corresponding author. Email:chunhai2001@163.com (C. H. Li)
Get access

Abstract

Dynamical system theory is applied to the integrable nonlinear wave equation ut ±(u 3–u 2)x +(u 3)xxx =0. We obtain the single peak solitary wave solutions and compacton solutions of the equation. Regular compacton solution of the equation correspond to the case of wave speed c=0. In the case of c≠0, we find smooth soliton solutions. The influence of parameters of the traveling wave solutions is explored by using the phase portrait analytical technique. Asymptotic analysis and numerical simulations are provided for these soliton solutions of the nonlinear wave equation.

Keywords

Bifurcationsolitary wavecompaction

MSC classification

Secondary: 35Q51: Soliton-like equations 35Q53: KdV-like equations (Korteweg-de Vries)

Information

Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 8 , Issue 6 , December 2016 , pp. 1084 - 1098
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1] Wazwaz, A., New soliton and kink solutions for the Gardner equation, Commum. Nonlinear Sci. Numer. Simulat., 12 (2007), pp. 1395–1404.CrossRefGoogle Scholar
[2] Khalique, C. M. and Biswas, A., A Lie symmetry approach to nonlinear Schrödinger's equation with non-Kerr law nonlinearity, Commum. Nonlinear Sci. Numer. Simulat., 14 (2009), pp. 4033–4040.CrossRefGoogle Scholar
[3] Chen, A. Y., Li, J. B. and Huang, W. T., The monotonicity and critical periods of periodic waves of the ϕ6 field model, Nonlinear Dyn., 63 (2011), pp. 205–215.Google Scholar
[4] Zhou, J. B., Tian, L. X. and Fan, X. H., New exact travelling wave solutions for the equation with Osmosis dispersion, Appl. Math. Comput., 217 (2010), pp. 1355–1366.Google Scholar
[5] Rui, W. G., Different kinds of exact solutions with two-loop character of the two-component short pulse equations of the first kind, Commun. Nonlinear Sci. Numer. Simulat., 18 (2013), pp. 2667–2678.Google Scholar
[6] Chen, A. Y. and Li, J. B., Single peak solitary wave solutions for the osmosis K(2,2) equation under inhomogeneous houndary condition, J. Math. Anal. Appl., 369 (2010), pp.758–766.Google Scholar
[7] Li, J. B. and Dai, H. H., Singular Nonlinear Traveling Wave Equations: Bifurcations and Exact Solutions, Science Press, Beijing, 2007.Google Scholar
[8] Li, J. B. and Chen, G. R., On a class of singular nonlinear traveling wave equations, Int. J. Bifurcation Chaos., 17 (2007), pp. 4049–4065.Google Scholar
[9] Li, J. B., Zhang, Y. and Zhao, X. H., On a class of singular nonlinear traveling wave equations, (II): an example of GCKdv equations, Int. J. Bifurcation Chaos., 19 (2009), pp. 1995–2007.Google Scholar
[10] Camassa, R. and Holm, D., An integrable shallow water equation with peaked solution, Phys. Rev. Lett., 71 (1993), pp. 1161–1164.Google Scholar
[11] Lenells, J., Traveling wave solutions of the Camassa-Holm equation, J. Diff. Equ., 217 (2005), pp. 393–430.Google Scholar
[12] Chen, A. Y., Wen, S. Q., Tang, S. Q., Huang, W. T. and Qiao, Z. J., Effects of quadratic singular curves in integrable equations, Stud. Appl. Math., 134 (2015), pp. 24–61.Google Scholar
[13] Qiao, Z. J. and Zhang, G. P., On peaked and smooth solitons for the Camassa-Holm equation, Europhy. Lett., 73 (2006), pp. 657–663.Google Scholar
[14] Zhu, W. J., Chen, A. Y. and Liu, Q. H., Periodic wave, solitary wave and compacton solutions of a nonlinear wave equation with degenerate dispersion, Commun. Theor. Phys., 63 (2015), pp. 57–62.Google Scholar
[15] Li, C. H., Wen, S. Q. and Chen, A. Y., Single peak solitary wave and compacton solutions of the generalized two-component Hunter-Saxton system, Nolinear Dyn., 79 (2015), pp. 1575–1585.Google Scholar
[16] Zhang, L., Chen, L. and Hou, X., The effects of horizontal singular straight line in a generalized nonlinear Klein-Gordon model equation, Nonlinear Dyn., 72 (2013), pp. 789–801.Google Scholar
[17] Wang, Y. and Bi, Q., Different wave solutions associated with singular lines on phase plane, Nonlinear Dyn., 69 (2012), pp. 1705–1731.Google Scholar
[18] Rosenau, P., On nonanalytic solitary waves formed by a nonlinear dispersion, Phys. Lett. A, 230 (1997), pp. 305–318.Google Scholar
[19] Rosenau, P., Nonlinear dispersion and compact structures, Phys. Rev. Lett., 73 (1993), pp. 1737–1741.Google Scholar
[20] Rosenau, P. and Hyman, J. M., Staley, multidimensional compacton, Phys. Rev. Lett., 98 (2007), pp. 024101.Google Scholar
[21] Rosenau, P., Compactification of patterns by a singular convection or stress, Phys. Rev. Lett., 99 (2007), pp. 234102.Google Scholar
[22] Rosenau, P. and Kashdan, E., Emergence of compact structures in a Klein-Gordon model, Phys. Rev. Lett., 104 (2010), pp. 034101.Google Scholar
[23] Cooper, F., Shepard, H. and Sodano, P., Solitary wave in a class of generalized Korteweg-de Vries equations, Phys. Rev. E, 48 (1993), pp. 4027–4032.CrossRefGoogle Scholar
[24] Zhou, J. B. and Tian, L. X., Soliton solution of the osmosis K(2,2) equation, Phys. Lett. A, 372 (2008), pp. 6232–6234.Google Scholar
[25] Chen, A. Y., Zhu, W. J., Qiao, Z. J. and Huang, W. T., Algebraic traveling wave solutions of a non-local hydrodynamic-type model, Math. Phys. Anal. Geom., 17 (2014), pp. 465–482.Google Scholar
[26] Chen, A. Y., Huang, W. T. and Xie, Y. A., Nilpotent singular points and compactons, Appl. Math. Comput., 236 (2014), pp. 300–310.Google Scholar
[27] Rosenau, P. and Hyman, J. M., Compactons: solitons with finite wavelengths, Phys. Rev. Lett., 70 (1993), pp. 564–567.Google Scholar
[28] Xu, C. H. and Tian, L. X., The bifurcation and peakon for K(2,2) equation with osmosis dispersion, Chaos Solitons Fract., 40 (2009), pp. 893–901.Google Scholar
[29] Rosenau, P. and Oron, A., On compactons induced by a non-convex convection, Nonlinear Sci. Numer. Simulat., 19 (2014), pp. 1329–1337.Google Scholar