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 & 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. doi: 10.1023/A:1021186408422. Google Scholar

[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. Google Scholar

[3]

R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661. doi: 10.1103/PhysRevLett.71.1661. Google Scholar

[4]

F. Cooper and H. Shepard, Solitons in the Camassa-Holm shallow water equation,, Phys. Lett. A, 194 (1994), 246. doi: 10.1016/0375-9601(94)91246-7. Google Scholar

[5]

A. Constantin, Soliton interactions for the Camassa-Holm equation,, Exposition. Math., 15 (1997), 251. Google Scholar

[6]

A. Constantin and W. A. Strauss, Stability of Peakons,, Comm. Pure Appl. Math., 53 (2000), 603. doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L. Google Scholar

[7]

J. P. Boyd, Peakons and coshoidal waves: travelling wave solutions of the Camassa-Holm equation,, Appl. Math. Comput., 81 (1997), 173. doi: http://dx.doi.org/10.1016/0096-3003(95)00326-6. Google Scholar

[8]

J. Lenells, The scattering approach for the Camassa-Holm equation,, J. Non. Math. Phys., 9 (2002), 389. doi: 10.2991/jnmp.2002.9.4.2. Google Scholar

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R. S. Johnson, Camassa-Holm, Korteweg-de Vries and rlated models for water waves,, J. Fluid Mech., 455 (2002), 63. doi: 10.1017/S0022112001007224. Google Scholar

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E. G. Reyes, Geometric integrability of the Camassa-Holm equation,, Lett. Math. Phys., 59 (2002), 117. doi: 10.1023/A:1014933316169. Google Scholar

[11]

Z. R. Liu, R. Q. Wand and Z. J. Jing, Peaked wave solutions of Camassa-Holm equation,, Chaos Solitons Fract., 19 (2004), 77. doi: 10.1016/S0960-0779(03)00082-1. Google Scholar

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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. doi: 10.1142/S0218127406016045. Google Scholar

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A. Degasperis and M. Procesi, Asymptotic integrability,, in, (1999), 23. Google Scholar

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H. Lundmark and J. Szmigielski, Multi-peakon solutions of the Degasperis-Procesi equation,, Inverse Probl., 19 (2003), 1241. doi: 10.1088/0266-5611/19/6/001. Google Scholar

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H. Lundmark and J. Szmigielski, Degasperis-Procesi peakons and the discrete cubic string,, Int. Math. Res. Pap., 2 (2005), 53. doi: 10.1155/IMRP.2005.53. Google Scholar

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C. Chen and M. Y. Tang, A new type of bounded waves for Degasperis-Procesi equations,, Chaos Soliton Fract., 27 (2006), 698. doi: 10.1016/j.chaos.2005.04.040. Google Scholar

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P. Guha, Euler-Poincare formalism of (two component) Degasperis-Procesi and Holm-Staley type systems,, J. Non. Math. Phys., 14 (2007), 390. doi: 10.2991/jnmp.2007.14.3.8. Google Scholar

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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. doi: 10.1016/S0375-9601(03)00114-2. Google Scholar

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B. L. Guo and Z. R. Liu, Periodic cusp wave solutions and single-solitons for the $b$-equation,, Chaos Soliton Fract., 23 (2005), 1451. doi: 10.1016/j.chaos.2004.06.062. Google Scholar

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Z. R. Liu and T. F. Qian, Peakons and their bifurcation in a generalized Camassa-Holm equation,, Int. J. Bifurcat. Chaos, 11 (2001), 781. doi: 10.1142/S0218127401002420. Google Scholar

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A. M. Wazwaz, Solitary wave solutions for modified forms of Degasperis-Procesi and Camassa-Holm equations,, Phys. Lett. A, 352 (2006), 500. doi: 10.1016/j.physleta.2005.12.036. Google Scholar

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A. M. Wazwaz, New solitary wave solutions to the modified forms of Degasperis-Procesi and Camassa-Holm equations,, Appl. Math. Comput., 186 (2007), 130. doi: 10.1016/j.amc.2006.07.092. Google Scholar

[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. doi: 10.1016/S0960-0779(03)00192-9. Google Scholar

[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. doi: 10.1016/j.chaos.2005.02.021. Google Scholar

[25]

S. A. Khuri, New ansatz for obtaining wave solutions of the generalized Camassa-Holm equation,, Chaos Soliton Fract., 25 (2005), 705. doi: 10.1016/j.chaos.2004.11.083. Google Scholar

[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. doi: 10.1016/j.physleta.2007.01.074. Google Scholar

[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. doi: 10.1016/j.amc.2008.08.043. Google Scholar

[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. doi: 10.1016/j.pnsc.2007.11.004. Google Scholar

[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. doi: 10.1016/j.cam.2006.04.047. Google Scholar

[30]

Q. D. Wang and M. Y. Tang, New exact solutions for two nonlinear equations,, Phys. Lett. A, 372 (2008), 2995. doi: 10.1016/j.physleta.2008.01.012. Google Scholar

[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. doi: 10.1016/j.physleta.2007.03.008. Google Scholar

[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. doi: 10.1016/j.physleta.2007.09.003. Google Scholar

[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. doi: 10.1016/j.amc.2008.08.042. Google Scholar

[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. doi: 10.1142/S0218127409024050. Google Scholar

[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. doi: 10.3934/cpaa.2010.9.77. Google Scholar

show all references

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. doi: 10.1023/A:1021186408422. Google Scholar

[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. Google Scholar

[3]

R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661. doi: 10.1103/PhysRevLett.71.1661. Google Scholar

[4]

F. Cooper and H. Shepard, Solitons in the Camassa-Holm shallow water equation,, Phys. Lett. A, 194 (1994), 246. doi: 10.1016/0375-9601(94)91246-7. Google Scholar

[5]

A. Constantin, Soliton interactions for the Camassa-Holm equation,, Exposition. Math., 15 (1997), 251. Google Scholar

[6]

A. Constantin and W. A. Strauss, Stability of Peakons,, Comm. Pure Appl. Math., 53 (2000), 603. doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L. Google Scholar

[7]

J. P. Boyd, Peakons and coshoidal waves: travelling wave solutions of the Camassa-Holm equation,, Appl. Math. Comput., 81 (1997), 173. doi: http://dx.doi.org/10.1016/0096-3003(95)00326-6. Google Scholar

[8]

J. Lenells, The scattering approach for the Camassa-Holm equation,, J. Non. Math. Phys., 9 (2002), 389. doi: 10.2991/jnmp.2002.9.4.2. Google Scholar

[9]

R. S. Johnson, Camassa-Holm, Korteweg-de Vries and rlated models for water waves,, J. Fluid Mech., 455 (2002), 63. doi: 10.1017/S0022112001007224. Google Scholar

[10]

E. G. Reyes, Geometric integrability of the Camassa-Holm equation,, Lett. Math. Phys., 59 (2002), 117. doi: 10.1023/A:1014933316169. Google Scholar

[11]

Z. R. Liu, R. Q. Wand and Z. J. Jing, Peaked wave solutions of Camassa-Holm equation,, Chaos Solitons Fract., 19 (2004), 77. doi: 10.1016/S0960-0779(03)00082-1. Google Scholar

[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. doi: 10.1142/S0218127406016045. Google Scholar

[13]

A. Degasperis and M. Procesi, Asymptotic integrability,, in, (1999), 23. Google Scholar

[14]

H. Lundmark and J. Szmigielski, Multi-peakon solutions of the Degasperis-Procesi equation,, Inverse Probl., 19 (2003), 1241. doi: 10.1088/0266-5611/19/6/001. Google Scholar

[15]

H. Lundmark and J. Szmigielski, Degasperis-Procesi peakons and the discrete cubic string,, Int. Math. Res. Pap., 2 (2005), 53. doi: 10.1155/IMRP.2005.53. Google Scholar

[16]

C. Chen and M. Y. Tang, A new type of bounded waves for Degasperis-Procesi equations,, Chaos Soliton Fract., 27 (2006), 698. doi: 10.1016/j.chaos.2005.04.040. Google Scholar

[17]

P. Guha, Euler-Poincare formalism of (two component) Degasperis-Procesi and Holm-Staley type systems,, J. Non. Math. Phys., 14 (2007), 390. doi: 10.2991/jnmp.2007.14.3.8. Google Scholar

[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. doi: 10.1016/S0375-9601(03)00114-2. Google Scholar

[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. doi: 10.1016/j.chaos.2004.06.062. Google Scholar

[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. doi: 10.1142/S0218127401002420. Google Scholar

[21]

A. M. Wazwaz, Solitary wave solutions for modified forms of Degasperis-Procesi and Camassa-Holm equations,, Phys. Lett. A, 352 (2006), 500. doi: 10.1016/j.physleta.2005.12.036. Google Scholar

[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. doi: 10.1016/j.amc.2006.07.092. Google Scholar

[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. doi: 10.1016/S0960-0779(03)00192-9. Google Scholar

[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. doi: 10.1016/j.chaos.2005.02.021. Google Scholar

[25]

S. A. Khuri, New ansatz for obtaining wave solutions of the generalized Camassa-Holm equation,, Chaos Soliton Fract., 25 (2005), 705. doi: 10.1016/j.chaos.2004.11.083. Google Scholar

[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. doi: 10.1016/j.physleta.2007.01.074. Google Scholar

[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. doi: 10.1016/j.amc.2008.08.043. Google Scholar

[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. doi: 10.1016/j.pnsc.2007.11.004. Google Scholar

[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. doi: 10.1016/j.cam.2006.04.047. Google Scholar

[30]

Q. D. Wang and M. Y. Tang, New exact solutions for two nonlinear equations,, Phys. Lett. A, 372 (2008), 2995. doi: 10.1016/j.physleta.2008.01.012. Google Scholar

[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. doi: 10.1016/j.physleta.2007.03.008. Google Scholar

[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. doi: 10.1016/j.physleta.2007.09.003. Google Scholar

[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. doi: 10.1016/j.amc.2008.08.042. Google Scholar

[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. doi: 10.1142/S0218127409024050. Google Scholar

[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. doi: 10.3934/cpaa.2010.9.77. Google Scholar

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