BLOW-UP SOLUTIONS FOR A CASE OF b-FAMILY EQUATIONS

doi:10.1007/s10473-020-0402-4

数学物理学报(英文版) ›› 2020, Vol. 40 ›› Issue (4): 910-920.doi: 10.1007/s10473-020-0402-4

BLOW-UP SOLUTIONS FOR A CASE OF b-FAMILY EQUATIONS

李宗广, 刘锐

School of Mathematics, South China University of Technology, Guangzhou 510640, China

收稿日期:2019-06-24 修回日期:2019-12-28 出版日期:2020-08-25 发布日期:2020-08-21
通讯作者: Rui LIU E-mail:scliurui@scut.edu.cn
作者简介:Zongguang LI,E-mail:337303753@qq.com
基金资助:
This research was partially supported by the National Natural Science Foundation of China (11771152, 11971176), Guangdong Basic and Applied Basic Research Foundation (2019B151502062), and the Fundamental Research Founds for the Central Universities (2019MS111).

BLOW-UP SOLUTIONS FOR A CASE OF b-FAMILY EQUATIONS

Zongguang LI, Rui LIU

School of Mathematics, South China University of Technology, Guangzhou 510640, China

Received:2019-06-24 Revised:2019-12-28 Online:2020-08-25 Published:2020-08-21
Contact: Rui LIU E-mail:scliurui@scut.edu.cn
Supported by:
This research was partially supported by the National Natural Science Foundation of China (11771152, 11971176), Guangdong Basic and Applied Basic Research Foundation (2019B151502062), and the Fundamental Research Founds for the Central Universities (2019MS111).

摘要/Abstract

摘要： In this article, we study the blow-up solutions for a case of b-family equations. Using the qualitative theory of differential equations and the bifurcation method of dynamical systems, we obtain five types of blow-up solutions: the hyperbolic blow-up solution, the fractional blow-up solution, the trigonometric blow-up solution, the first elliptic blow-up solution, and the second elliptic blow-up solution. Not only are the expressions of these blow-up solutions given, but also their relationships are discovered. In particular, it is found that two bounded solitary solutions are bifurcated from an elliptic blow-up solution.

关键词: b-family equation, blow-up solutions, qualitative theory, bifurcation method

Abstract: In this article, we study the blow-up solutions for a case of b-family equations. Using the qualitative theory of differential equations and the bifurcation method of dynamical systems, we obtain five types of blow-up solutions: the hyperbolic blow-up solution, the fractional blow-up solution, the trigonometric blow-up solution, the first elliptic blow-up solution, and the second elliptic blow-up solution. Not only are the expressions of these blow-up solutions given, but also their relationships are discovered. In particular, it is found that two bounded solitary solutions are bifurcated from an elliptic blow-up solution.

Key words: b-family equation, blow-up solutions, qualitative theory, bifurcation method

中图分类号:

34A20

李宗广, 刘锐. BLOW-UP SOLUTIONS FOR A CASE OF b-FAMILY EQUATIONS[J]. 数学物理学报(英文版), 2020, 40(4): 910-920.

Zongguang LI, Rui LIU. BLOW-UP SOLUTIONS FOR A CASE OF b-FAMILY EQUATIONS[J]. Acta mathematica scientia,Series B, 2020, 40(4): 910-920.

参考文献

[1] Camassa R, Holm D D. An integrable shallow water equation with peaked solitons. Phys Rev Lett, 1993, 71:1661-1664
[2] Degasperis A, Procesi M. Asymptotic integrability//Degasperis A, Gaeta G, eds. Symmetry and Perturbation Theory. Singapore:World Sci Publishing. 1999:23-37
[3] Constantin A, Strauss W A. Stability of Peakons. Commun Pur Appl Math, 2000, 53:603-610
[4] Johnson R S. Camassa-Holm, Korteweg-de Vries and related models for water waves. J Fluid Mech, 2002, 455:63-82
[5] Lundmark H, Szmigielski J. Degasperis-Procesi peakons and the discrete cubic string. Int Math Res Papers, 2005, (2):53-116
[6] Wu S, Yin Z. Blow-up and decay of the solution ofthe weakly dissipative Degasperis-Procesi equation. SIAM J Math Anal, 2008, 40:475-490
[7] Escher J, Kolev B. The Degasperis-Procesi equation as a non-metric Eulet equation. Math Z, 2011, 269:1137-1153
[8] Zhang L, Liu B. The global attractor for a viscous weakly dissipative generalized two-componeut μ-Hunter-Saxton system. Acta Math Sci, 2018, 38B(2):651-672
[9] Tian L X, Song X Y. New peaked solitary wave solutions of the generalized Camassa-Holm equation. Chaos Solit Fract, 2004, 21:621-637
[10] Shen J W, Xu W. Bifurcations of smooth and non-smooth travelling wave solutions in the generalized Camassa-Holm equation. Chaos Solit Fract, 2005, 26:1149-1162
[11] Khuri S A. New ansatz for obtaining wave solutions of the generalized Camassa-Holm equation. Chaos Solit. Fract. 2005, 25:705-710
[12] Wazwaz A M. New solitary wave solutions to the modied forms of Degasperis-Procesi and Camassa-Holm equations. Appl Math Comput, 2007, 186:130-141
[13] He B, Rui W G, Li S L, et al. Bounded travelling wave solutions for a modified form of generalized Degasperis-Procesi equation. Appl Math Comput, 2008, 206:113-123
[14] Liu Z R, Tang H. Explicit periodic wave solutions and their bifurcations for generalized Camassa-Holm equation. Int J Bifurcat Chaos, 2010, 20(8):2507-2519
[15] Liu Z R, Liang Y. The explicit nonlinear wave solutions and their bifurcations of the generalized Camassa-Holm equation. Int J Bifurcat Chaos, 2011, 21:3119-3136
[16] Liu R. Coexistence of multifarious exact nonlinear wave solutions for generalized b-equation. Int J Bifurcat Chaos, 2010, 20:3193-3208
[17] Liu R. The explicit nonlinear wave solutions of the generalized b-equation. Commun Pure Appl Anal, 2013, 12(2):1029-1047
[18] Chen Y R, Ye W B, Liu R. The explicit periodic wave solutions and their limit forms for a generalized b-equation. Acta Mathematicae Applicatae Sinica, 2016, 32(2):513-528
[19] Yang J P, Li Z G, Liu Z R. The existence and bifurcation of peakon and anti-peakon to the n-degree b-equation. Int J Bifurcat Chaos, 2018, 28(1):1850014
[20] Li Z G, Liu R. Bifurcations and exact solutions in a nonlinear wave equation. Int J Bifurcat Chaos, 2019, 29(7):1950098
[21] Yang Z, Zhang G. Global stability of traveling wave fronts for nonlocal reacton-diffusion equations with time delay. Acta Math Sci, 2018, 38B(1):289-302
[22] Zhang W G, Li W X, Deng S E, Li X. Asymptotic stability of monotone decreasing Kink profile solitary wave solutions for generalized KdV-Burgers equation. Acta Math Appl Sci, Engl Series, 2019, 35B(3):475-490

BLOW-UP SOLUTIONS FOR A CASE OF b-FAMILY EQUATIONS

BLOW-UP SOLUTIONS FOR A CASE OF b-FAMILY EQUATIONS

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