# American Institute of Mathematical Sciences

March  2013, 12(2): 1029-1047. doi: 10.3934/cpaa.2013.12.1029

## The explicit nonlinear wave solutions of the generalized $b$-equation

 1 Department of Mathematics, South China University of Technology, Guangzhou 510640, China

Received  January 2011 Revised  July 2012 Published  September 2012

In this paper, we study the nonlinear wave solutions of the generalized $b$-equation involving two parameters $b$ and $k$. Let $c$ be constant wave speed, $c_5=$ $\frac{1}{2}(1+b-\sqrt{(1+b)(1+b-8k)})$, $c_6=\frac{1}{2}(1+b+\sqrt{(1+b)(1+b-8k)})$. We obtain the following results:

1. If $-\infty < k < \frac{1+b}{8}$ and $c\in (c_5, c_6)$, then there are three types of explicit nonlinear wave solutions, hyperbolic smooth solitary wave solution, hyperbolic peakon wave solution and hyperbolic blow-up solution.

2. If $-\infty < k < \frac{1+b}{8}$ and $c=c_5$ or $c_6$, then there are two types of explicit nonlinear wave solutions, fractional peakon wave solution and fractional blow-up solution.

3. If $k=\frac{1+b}{8}$ and $c=\frac{b+1}{2}$, then there are two types of explicit nonlinear wave solutions, fractional peakon wave solution and fractional blow-up solution.

Not only is the existence of these solutions shown, but their concrete expressions are presented. We also reveal the relationships among these solutions. Besides, the correctness of these solutions is tested by using the software Mathematica.
Citation: Liu Rui. The explicit nonlinear wave solutions of the generalized $b$-equation. Communications on Pure and Applied Analysis, 2013, 12 (2) : 1029-1047. doi: 10.3934/cpaa.2013.12.1029
##### References:
 [1] A. Degasperis, D. D. Holm and A. N. W. Hone, A new integrable equation with peakon solutions, Theoret. and Math. Phys., 133 (2002), 1463-1474. doi: 10.1023/A:1021186408422. [2] A. Degasperis, D. D. Holm and A. N. W. Hone, Integrable and non-integrable equations with peakons, Nonlinear physics: Theory and experiment, II (2002), 37-43. World Sci Publishing, River Edge, NJ, 2003. [3] R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661. [4] F. Cooper and H. Shepard, Solitons in the Camassa-Holm shallow water equation, Phys. Lett. A, 194 (1994), 246-250. doi: 10.1016/0375-9601(94)91246-7. [5] A. Constantin, Soliton interactions for the Camassa-Holm equation, Exposition. Math., 15 (1997), 251-264. [6] A. Constantin and W. A. Strauss, Stability of Peakons, Comm. Pure Appl. Math., 53 (2000), 603-610. doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L. [7] J. P. Boyd, Peakons and coshoidal waves: travelling wave solutions of the Camassa-Holm equation, Appl. Math. Comput., 81 (1997), 173-187. doi: http://dx.doi.org/10.1016/0096-3003(95)00326-6. [8] J. Lenells, The scattering approach for the Camassa-Holm equation, J. Non. Math. Phys., 9 (2002), 389-393. doi: 10.2991/jnmp.2002.9.4.2. [9] R. S. Johnson, Camassa-Holm, Korteweg-de Vries and rlated models for water waves, J. Fluid Mech., 455 (2002), 63-82. doi: 10.1017/S0022112001007224. [10] E. G. Reyes, Geometric integrability of the Camassa-Holm equation, Lett. Math. Phys., 59 (2002), 117-131. doi: 10.1023/A:1014933316169. [11] Z. R. Liu, R. Q. Wand and Z. J. Jing, Peaked wave solutions of Camassa-Holm equation, Chaos Solitons Fract., 19 (2004), 77-92. doi: 10.1016/S0960-0779(03)00082-1. [12] Z. R. Liu, A. M. Kayed and C. Chen, Periodic waves and their limits for the Camassa-Holm equation, Int. J. Bifurcat. Chaos, 16 (2006), 2261-2274. doi: 10.1142/S0218127406016045. [13] A. Degasperis and M. Procesi, Asymptotic integrability, in "Symmetry and Perturbation Theory" (eds. A.Degasperis and G.Gaeta), World Sci Publishing, (1999), 23-37. [14] H. Lundmark and J. Szmigielski, Multi-peakon solutions of the Degasperis-Procesi equation, Inverse Probl., 19 (2003), 1241-1245. doi: 10.1088/0266-5611/19/6/001. [15] H. Lundmark and J. Szmigielski, Degasperis-Procesi peakons and the discrete cubic string, Int. Math. Res. Pap., 2 (2005), 53-116. doi: 10.1155/IMRP.2005.53. [16] C. Chen and M. Y. Tang, A new type of bounded waves for Degasperis-Procesi equations, Chaos Soliton Fract., 27 (2006), 698-704. doi: 10.1016/j.chaos.2005.04.040. [17] P. Guha, Euler-Poincare formalism of (two component) Degasperis-Procesi and Holm-Staley type systems, J. Non. Math. Phys., 14 (2007), 390-421. doi: 10.2991/jnmp.2007.14.3.8. [18] D. D. Holm and M. F. Staley, Nonlinear balance and exchange of stability in dynamics of solitons, peakons ramps/cliffs and leftons in a $1+1$ nolinear evolutionary PDEs, Phys. Lett. A., 308 (2003), 437-444. doi: 10.1016/S0375-9601(03)00114-2. [19] B. L. Guo and Z. R. Liu, Periodic cusp wave solutions and single-solitons for the $b$-equation, Chaos Soliton Fract., 23 (2005), 1451-1463. doi: 10.1016/j.chaos.2004.06.062. [20] Z. R. Liu and T. F. Qian, Peakons and their bifurcation in a generalized Camassa-Holm equation, Int. J. Bifurcat. Chaos, 11 (2001), 781-792. doi: 10.1142/S0218127401002420. [21] A. M. Wazwaz, Solitary wave solutions for modified forms of Degasperis-Procesi and Camassa-Holm equations, Phys. Lett. A, 352 (2006), 500-504. doi: 10.1016/j.physleta.2005.12.036. [22] A. M. Wazwaz, New solitary wave solutions to the modified forms of Degasperis-Procesi and Camassa-Holm equations, Appl. Math. Comput., 186 (2007), 130-141. doi: 10.1016/j.amc.2006.07.092. [23] L. X. Tian and X. Y. Song, New peaked solitary wave solutions of the generalized Camassa-Holm equation, Chaos Soliton Fract., 21 (2004), 621-637. doi: 10.1016/S0960-0779(03)00192-9. [24] J. W. Shen and W. Xu, Bifurcations of smooth and non-smooth travelling wave solutions in the generalized Camassa-Holm equation, Chaos Soliton Fract., 26 (2005), 1149-1162. doi: 10.1016/j.chaos.2005.02.021. [25] S. A. Khuri, New ansatz for obtaining wave solutions of the generalized Camassa-Holm equation, Chaos Soliton Fract., 25 (2005), 705-710. doi: 10.1016/j.chaos.2004.11.083. [26] Z. R. Liu and Z. Y. Ouyang, A note on solitary waves for modified forms of Camassa-Holm and Degasperis-Procesi equations, Phys. Lett. A, 366 (2007), 377-381. doi: 10.1016/j.physleta.2007.01.074. [27] B. He, W. G. Rui and C. Chen, Exact travelling wave solutions for a generalized Camassa-Holm equation using the integral bifurcation method, Appl. Math. Comput., 206 (2008), 141-149. doi: 10.1016/j.amc.2008.08.043. [28] Z. R. Liu and B. L. Guo, Periodic blow-up solutions and their limit forms for the generalized Camassa-Holm equation, Prog. Nat. Sci., 18 (2008), 259-266. doi: 10.1016/j.pnsc.2007.11.004. [29] L. J. Zhang, Q. C. Li and X. W. Huo, Bifurcations of smooth and nonsmooth travelling wave solutions in a generalized Degasperis-Procesi equation, J. Comput. Appl. Math., 205 (2007), 174-185. doi: 10.1016/j.cam.2006.04.047. [30] Q. D. Wang and M. Y. Tang, New exact solutions for two nonlinear equations, Phys. Lett. A, 372 (2008), 2995-3000. doi: 10.1016/j.physleta.2008.01.012. [31] E. Yomba, The sub-ODE method for finding exact travelling wave solutions of generalized nonlinear Camassa-Holm, and generalized nonlinear Schrödinger equations, Phys. Lett. A, 372 (2008), 215-222. doi: 10.1016/j.physleta.2007.03.008. [32] E. Yomba, A generalized auxiliary equation method and its application to nonlinear Klein-Gordon and generalized nonlinear Camassa-Holm equations, Phys. Lett. A, 372 (2008), 1048-1060. doi: 10.1016/j.physleta.2007.09.003. [33] B. He, W. G. Rui and S. L. Li, Bounded travelling wave solutions for a modified form of generalized Degasperis-Procesi equation, Appl. Math. Comput., 206 (2008), 113-123. doi: 10.1016/j.amc.2008.08.042. [34] Z. R. Liu and J. Pan, Coexistence of multifarious explicit nonlinear wave solutions for modified forms of Camassa-Holm and Degaperis-Procesi equations, Int. J. Bifurcat. Chaos, 19 (2009), 2267-2282. doi: 10.1142/S0218127409024050. [35] R. Liu, Several new types of solitary wave solutions for the generalized Camassa-Holm-Degasperis-Procesi equation, Commun. Pur. Appl. Anal., 9 (2010), 77-90. doi: 10.3934/cpaa.2010.9.77.

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##### References:
 [1] A. Degasperis, D. D. Holm and A. N. W. Hone, A new integrable equation with peakon solutions, Theoret. and Math. Phys., 133 (2002), 1463-1474. doi: 10.1023/A:1021186408422. [2] A. Degasperis, D. D. Holm and A. N. W. Hone, Integrable and non-integrable equations with peakons, Nonlinear physics: Theory and experiment, II (2002), 37-43. World Sci Publishing, River Edge, NJ, 2003. [3] R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661. [4] F. Cooper and H. Shepard, Solitons in the Camassa-Holm shallow water equation, Phys. Lett. A, 194 (1994), 246-250. doi: 10.1016/0375-9601(94)91246-7. [5] A. Constantin, Soliton interactions for the Camassa-Holm equation, Exposition. Math., 15 (1997), 251-264. [6] A. Constantin and W. A. Strauss, Stability of Peakons, Comm. Pure Appl. Math., 53 (2000), 603-610. doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L. [7] J. P. Boyd, Peakons and coshoidal waves: travelling wave solutions of the Camassa-Holm equation, Appl. Math. Comput., 81 (1997), 173-187. doi: http://dx.doi.org/10.1016/0096-3003(95)00326-6. [8] J. Lenells, The scattering approach for the Camassa-Holm equation, J. Non. Math. Phys., 9 (2002), 389-393. doi: 10.2991/jnmp.2002.9.4.2. [9] R. S. Johnson, Camassa-Holm, Korteweg-de Vries and rlated models for water waves, J. Fluid Mech., 455 (2002), 63-82. doi: 10.1017/S0022112001007224. [10] E. G. Reyes, Geometric integrability of the Camassa-Holm equation, Lett. Math. Phys., 59 (2002), 117-131. doi: 10.1023/A:1014933316169. [11] Z. R. Liu, R. Q. Wand and Z. J. Jing, Peaked wave solutions of Camassa-Holm equation, Chaos Solitons Fract., 19 (2004), 77-92. doi: 10.1016/S0960-0779(03)00082-1. [12] Z. R. Liu, A. M. Kayed and C. Chen, Periodic waves and their limits for the Camassa-Holm equation, Int. J. Bifurcat. Chaos, 16 (2006), 2261-2274. doi: 10.1142/S0218127406016045. [13] A. Degasperis and M. Procesi, Asymptotic integrability, in "Symmetry and Perturbation Theory" (eds. A.Degasperis and G.Gaeta), World Sci Publishing, (1999), 23-37. [14] H. Lundmark and J. Szmigielski, Multi-peakon solutions of the Degasperis-Procesi equation, Inverse Probl., 19 (2003), 1241-1245. doi: 10.1088/0266-5611/19/6/001. [15] H. Lundmark and J. Szmigielski, Degasperis-Procesi peakons and the discrete cubic string, Int. Math. Res. Pap., 2 (2005), 53-116. doi: 10.1155/IMRP.2005.53. [16] C. Chen and M. Y. Tang, A new type of bounded waves for Degasperis-Procesi equations, Chaos Soliton Fract., 27 (2006), 698-704. doi: 10.1016/j.chaos.2005.04.040. [17] P. Guha, Euler-Poincare formalism of (two component) Degasperis-Procesi and Holm-Staley type systems, J. Non. Math. Phys., 14 (2007), 390-421. doi: 10.2991/jnmp.2007.14.3.8. [18] D. D. Holm and M. F. Staley, Nonlinear balance and exchange of stability in dynamics of solitons, peakons ramps/cliffs and leftons in a $1+1$ nolinear evolutionary PDEs, Phys. Lett. A., 308 (2003), 437-444. doi: 10.1016/S0375-9601(03)00114-2. [19] B. L. Guo and Z. R. Liu, Periodic cusp wave solutions and single-solitons for the $b$-equation, Chaos Soliton Fract., 23 (2005), 1451-1463. doi: 10.1016/j.chaos.2004.06.062. [20] Z. R. Liu and T. F. Qian, Peakons and their bifurcation in a generalized Camassa-Holm equation, Int. J. Bifurcat. Chaos, 11 (2001), 781-792. doi: 10.1142/S0218127401002420. [21] A. M. Wazwaz, Solitary wave solutions for modified forms of Degasperis-Procesi and Camassa-Holm equations, Phys. Lett. A, 352 (2006), 500-504. doi: 10.1016/j.physleta.2005.12.036. [22] A. M. Wazwaz, New solitary wave solutions to the modified forms of Degasperis-Procesi and Camassa-Holm equations, Appl. Math. Comput., 186 (2007), 130-141. doi: 10.1016/j.amc.2006.07.092. [23] L. X. Tian and X. Y. Song, New peaked solitary wave solutions of the generalized Camassa-Holm equation, Chaos Soliton Fract., 21 (2004), 621-637. doi: 10.1016/S0960-0779(03)00192-9. [24] J. W. Shen and W. Xu, Bifurcations of smooth and non-smooth travelling wave solutions in the generalized Camassa-Holm equation, Chaos Soliton Fract., 26 (2005), 1149-1162. doi: 10.1016/j.chaos.2005.02.021. [25] S. A. Khuri, New ansatz for obtaining wave solutions of the generalized Camassa-Holm equation, Chaos Soliton Fract., 25 (2005), 705-710. doi: 10.1016/j.chaos.2004.11.083. [26] Z. R. Liu and Z. Y. Ouyang, A note on solitary waves for modified forms of Camassa-Holm and Degasperis-Procesi equations, Phys. Lett. A, 366 (2007), 377-381. doi: 10.1016/j.physleta.2007.01.074. [27] B. He, W. G. Rui and C. Chen, Exact travelling wave solutions for a generalized Camassa-Holm equation using the integral bifurcation method, Appl. Math. Comput., 206 (2008), 141-149. doi: 10.1016/j.amc.2008.08.043. [28] Z. R. Liu and B. L. Guo, Periodic blow-up solutions and their limit forms for the generalized Camassa-Holm equation, Prog. Nat. Sci., 18 (2008), 259-266. doi: 10.1016/j.pnsc.2007.11.004. [29] L. J. Zhang, Q. C. Li and X. W. Huo, Bifurcations of smooth and nonsmooth travelling wave solutions in a generalized Degasperis-Procesi equation, J. Comput. Appl. Math., 205 (2007), 174-185. doi: 10.1016/j.cam.2006.04.047. [30] Q. D. Wang and M. Y. Tang, New exact solutions for two nonlinear equations, Phys. Lett. A, 372 (2008), 2995-3000. doi: 10.1016/j.physleta.2008.01.012. [31] E. Yomba, The sub-ODE method for finding exact travelling wave solutions of generalized nonlinear Camassa-Holm, and generalized nonlinear Schrödinger equations, Phys. Lett. A, 372 (2008), 215-222. doi: 10.1016/j.physleta.2007.03.008. [32] E. Yomba, A generalized auxiliary equation method and its application to nonlinear Klein-Gordon and generalized nonlinear Camassa-Holm equations, Phys. Lett. A, 372 (2008), 1048-1060. doi: 10.1016/j.physleta.2007.09.003. [33] B. He, W. G. Rui and S. L. Li, Bounded travelling wave solutions for a modified form of generalized Degasperis-Procesi equation, Appl. Math. Comput., 206 (2008), 113-123. doi: 10.1016/j.amc.2008.08.042. [34] Z. R. Liu and J. Pan, Coexistence of multifarious explicit nonlinear wave solutions for modified forms of Camassa-Holm and Degaperis-Procesi equations, Int. J. Bifurcat. Chaos, 19 (2009), 2267-2282. doi: 10.1142/S0218127409024050. [35] R. Liu, Several new types of solitary wave solutions for the generalized Camassa-Holm-Degasperis-Procesi equation, Commun. Pur. Appl. Anal., 9 (2010), 77-90. doi: 10.3934/cpaa.2010.9.77.
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