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.
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
.
DOI: 10.1007/s10473-020-0402-4
[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
