American Institute of Mathematical Sciences

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

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

Liu Rui 1,

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.
Keywords: dynamical system, Explicit nonlinear wave solution, solitary, bifurcation method., generalized $b$-equation.
Mathematics Subject Classification: Primary: 34A20, 34C35, 35B65; Secondary: 58F05, 76B2.
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

[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

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

[1]

Út V. Lê. Regularity of the solution of a nonlinear wave equation. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1099-1115. doi: 10.3934/cpaa.2010.9.1099
[2]

Y. A. Li, P. J. Olver. Convergence of solitary-wave solutions in a perturbed bi-Hamiltonian dynamical system I. Compactions and peakons. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 419-432. doi: 10.3934/dcds.1997.3.419
[3]

Rui Liu. Several new types of solitary wave solutions for the generalized Camassa-Holm-Degasperis-Procesi equation. Communications on Pure & Applied Analysis, 2010, 9 (1) : 77-90. doi: 10.3934/cpaa.2010.9.77
[4]

Jong-Shenq Guo, Ying-Chih Lin. Traveling wave solution for a lattice dynamical system with convolution type nonlinearity. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 101-124. doi: 10.3934/dcds.2012.32.101
[5]

Jibin Li, Yi Zhang. On the traveling wave solutions for a nonlinear diffusion-convection equation: Dynamical system approach. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 1119-1138. doi: 10.3934/dcdsb.2010.14.1119
[6]

Yong Duan, Jian-Guo Liu. Convergence analysis of the vortex blob method for the $b$-equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1995-2011. doi: 10.3934/dcds.2014.34.1995
[7]

Zhaoquan Xu, Jiying Ma. Monotonicity, asymptotics and uniqueness of travelling wave solution of a non-local delayed lattice dynamical system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 5107-5131. doi: 10.3934/dcds.2015.35.5107
[8]

Y. A. Li, P. J. Olver. Convergence of solitary-wave solutions in a perturbed bi-hamiltonian dynamical system ii. complex analytic behavior and convergence to non-analytic solutions. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 159-191. doi: 10.3934/dcds.1998.4.159
[9]

Chui-Jie Wu, Hongliang Zhao. Generalized HWD-POD method and coupling low-dimensional dynamical system of turbulence. Conference Publications, 2001, 2001 (Special) : 371-379. doi: 10.3934/proc.2001.2001.371
[10]

Oana Pocovnicu. Explicit formula for the solution of the Szegö equation on the real line and applications. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 607-649. doi: 10.3934/dcds.2011.31.607
[11]

Amin Esfahani, Steve Levandosky. Solitary waves of the rotation-generalized Benjamin-Ono equation. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 663-700. doi: 10.3934/dcds.2013.33.663
[12]

Jerry L. Bona, Didier Pilod. Stability of solitary-wave solutions to the Hirota-Satsuma equation. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1391-1413. doi: 10.3934/dcds.2010.27.1391
[13]

Jibin Li, Yi Zhang. Exact solitary wave and quasi-periodic wave solutions for four fifth-order nonlinear wave equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 623-631. doi: 10.3934/dcdsb.2010.13.623
[14]

Jean-François Rault. A bifurcation for a generalized Burgers' equation in dimension one. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 683-706. doi: 10.3934/dcdss.2012.5.683
[15]

Jiao Chen, Weike Wang. The point-wise estimates for the solution of damped wave equation with nonlinear convection in multi-dimensional space. Communications on Pure & Applied Analysis, 2014, 13 (1) : 307-330. doi: 10.3934/cpaa.2014.13.307
[16]

Jibin Li. Family of nonlinear wave equations which yield loop solutions and solitary wave solutions. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 897-907. doi: 10.3934/dcds.2009.24.897
[17]

Tai-Chia Lin. Vortices for the nonlinear wave equation. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 391-398. doi: 10.3934/dcds.1999.5.391
[18]

Xinlong Feng, Huailing Song, Tao Tang, Jiang Yang. Nonlinear stability of the implicit-explicit methods for the Allen-Cahn equation. Inverse Problems & Imaging, 2013, 7 (3) : 679-695. doi: 10.3934/ipi.2013.7.679
[19]

Shaoyong Lai, Yong Hong Wu. The asymptotic solution of the Cauchy problem for a generalized Boussinesq equation. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 401-408. doi: 10.3934/dcdsb.2003.3.401
[20]

Saeed Ketabchi, Hossein Moosaei, M. Parandegan, Hamidreza Navidi. Computing minimum norm solution of linear systems of equations by the generalized Newton method. Numerical Algebra, Control & Optimization, 2017, 7 (2) : 113-119. doi: 10.3934/naco.2017008

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences