American Institute of Mathematical Sciences

Advanced
March  2013, 12(2): 1075-1090. doi: 10.3934/cpaa.2013.12.1075

Qualitative analysis to the traveling wave solutions of Kakutani-Kawahara equation and its approximate damped oscillatory solution

Weiguo Zhang 1, , Yan Zhao 1, and  Xiang Li 1,

1. 

School of Science, University of Shanghai for Science and Technology, Shanghai 200093, China, China, China

Received  August 2011 Revised  November 2012 Published  September 2012

In this paper, we apply the theory of planar dynamical systems to make a qualitative analysis to the traveling wave solutions of nonlinear Kakutani-Kawahara equation $u_t+uu_x+bu_{x x x}-a(u_t+uu_x)_x=0$ ($b>0, a\ge0$) and obtain the existent conditions of the bounded traveling wave solutions. In dispersion-dominant case, we find that the unique bounded traveling wave solution of this equation has not only oscillatory property but also damped property. Furthermore, according to the evolution of orbits in the global phase portraits, we present an approximate damped oscillatory solution for this equation by the undetermined coefficients method. Finally, by the idea of homogenization principles, we obtain an integral equation which reflects the relation between this approximate damped oscillatory solution and its exact solution, thereby having the error estimate. The error is an infinitesimal decreasing in exponential form. From the results in this paper, it can be seen that the damped oscillatory solution of Kakutani-Kawahara equation in dispersion-dominant case still remains some properties of solitary wave solution for KdV equation.
Keywords: qualitative analysis, error estimate., Nonlinear dispersive-dissipative equation, damped oscillatory solution, approximate solution.
Mathematics Subject Classification: Primary: 35Q53; Secondary: 35C0.
Citation: Weiguo Zhang, Yan Zhao, Xiang Li. Qualitative analysis to the traveling wave solutions of Kakutani-Kawahara equation and its approximate damped oscillatory solution. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1075-1090. doi: 10.3934/cpaa.2013.12.1075
References:
[1]

H. Washimi and T. Taniuti, Propagation of ion-acoustic solitary waves of small amplitude, Phys. Rev. Letters, 17 (1996), 996-998. doi: 10.1103/PhysRevLett.17.996.  Google Scholar

[2]

T. Kakutani and T. Kawahara, Weak ion-acoustic shock waves, J. Phys. Soc. Jpn., 29 (1970), 1068-1073. doi: 10.1143/JPSJ.29.1068.  Google Scholar

[3]

M. Malfliet, "The Tanh Method in Nonlinear Wave Theory, Habilitation Thesis," University of Antwerp, Antwerp, 1994. Google Scholar

[4]

N. Isidore, Exact solutions of a nonlinear dispersive-dissipative equation, J. Phys. A, 29 (1996), 3679-3682. doi: 10.1088/0305-4470/29/13/032.  Google Scholar

[5]

V. V. Nemytskii and V. V. Stepanov, "Qualitative Theory of Differential Equations," Princeton University Press, Princeton, 1989.  Google Scholar

[6]

Z. F. Zhang, T. R. Ding, W. Z. Huang and Z. X. Dong, "Qualitative Theory of Differential Equations," Translated from the Chinese by Anthony Wing Kwork Leung. Translations of Mathematical Monographs, 101, American Mathematical Society, Providence, RI, 1992.  Google Scholar

[7]

L. A. Cherkas, Estimation of the number of limit cycles of autonomous systems, Diff. Eqs., 13 (1977), 529-547. Google Scholar

[8]

J. B. Li and L. J. Zhang, Bifurcations of traveling wave solutions in generalized Pochhammer-Chree equation, Chaos. Soliton. Fract., 14 (2002), 581-593. doi: 10.1016/S0960-0779(01)00248-X.  Google Scholar

[9]

Q. X. Ye and Z. Y. Li, "Introduction of Reaction Diffusion Equations," Science Press, Beijing, 1990. (in Chinese)  Google Scholar

[10]

P. C. Fife, "Mathematical Aspects of Reacting and Diffusing Systems," Springer-Verlag, Berlin-New York, 1979.  Google Scholar

[11]

D. G. Aronson and H. F. Weiberger, Multidimentional nonlinear diffusion arising in population genetics, Adv.in Math., 30 (1978), 33-76. doi: doi:10.1016/0001-8708(78)90130-5.  Google Scholar

[12]

A. Jeffery and T. Kakutani, Stability of the Burgers shock wave and the Korteweg-de Vries soliton,, Indiana Univ. Math. J., 20 (): 463.  doi: 10.1512/iumj.1970.20.20039.  Google Scholar

[13]

J. L. Bona and M. E. Schonbek, Travelling wave solutions to the Korteweg-de Vries-Burgers equation, Proc. R. Soc. Edinburgh Sect. A, 101 (1985), 207-226. doi: 10.1017/S0308210500020783.  Google Scholar

[14]

H. Grad and P. N. Hu, Unified shock profile in a plasma, Phys. Fluids, 10 (1976), 2596-2602. doi: 10.1063/1.1762081.  Google Scholar

[15]

R. S. Johnson, A nonlinear equation incorporating damping and dispersion, J. Fluid Mech., 42 (1970), 49-60. doi: 10.1017/S0022112070001064.  Google Scholar

[16]

J. Canosa and J. Gazdag, The Korteweg-de Vries-Burgers equation, J. Computational Phys., 23 (1977), 393-403. doi: 0021-9991(77)90070-5.  Google Scholar

[17]

T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. R. Soc. London Ser. A, 272 (1972), 47-78. doi: 10.1098/rsta.1972.0032.  Google Scholar

[18]

J. L. Bona and V. A. Dougalis, An initial and boundary value problem for a model equation for propagation of long waves, J. Math. Anal. Appl., 75 (1980), 502-522. doi: 10.1016/0022-247X(80)90098-0.  Google Scholar

[19]

G. H. Ganser and D. A. Drew, Nonlinear periodic waves in a two-phase flow model, SIAM J. Appl. Math., 47 (1987), 726-736. doi: 10.1137/0147050.  Google Scholar

[20]

G. H. Ganser and D. A. Drew, Nonlinear stability analysis of a uniformly fluidized bed, Int. J. Multiphase Flow, 16 (1990), 447-460. doi: 10.1016/0301-9322(90)90075-T.  Google Scholar

[21]

L. Abia, I. Christie and J. M. Sanz-Serna, Stability of schemes for the numerical treatment of an equation modelling fluidized beds, Model. Math. Anal. Numer., 23 (1989), 191-204. doi: 0674.76022.  Google Scholar

[22]

J. C. Lopez-Marcos and J. M. Sanz-Serna, A definition of stability for nonlinear problems, Numerical Treatment of Differntial Equations, 104 (1988), 216-226.  Google Scholar

[23]

M. F. Feng and P. B. Ming, Stability finite difference method and the nonlinear stability analysis of modelling fluidized beds equation, Journal of Computational Mathematics in Colleges and Universities, 9 (1997), 298-311. Google Scholar

show all references

References:
[1]

H. Washimi and T. Taniuti, Propagation of ion-acoustic solitary waves of small amplitude, Phys. Rev. Letters, 17 (1996), 996-998. doi: 10.1103/PhysRevLett.17.996.  Google Scholar

[2]

T. Kakutani and T. Kawahara, Weak ion-acoustic shock waves, J. Phys. Soc. Jpn., 29 (1970), 1068-1073. doi: 10.1143/JPSJ.29.1068.  Google Scholar

[3]

M. Malfliet, "The Tanh Method in Nonlinear Wave Theory, Habilitation Thesis," University of Antwerp, Antwerp, 1994. Google Scholar

[4]

N. Isidore, Exact solutions of a nonlinear dispersive-dissipative equation, J. Phys. A, 29 (1996), 3679-3682. doi: 10.1088/0305-4470/29/13/032.  Google Scholar

[5]

V. V. Nemytskii and V. V. Stepanov, "Qualitative Theory of Differential Equations," Princeton University Press, Princeton, 1989.  Google Scholar

[6]

Z. F. Zhang, T. R. Ding, W. Z. Huang and Z. X. Dong, "Qualitative Theory of Differential Equations," Translated from the Chinese by Anthony Wing Kwork Leung. Translations of Mathematical Monographs, 101, American Mathematical Society, Providence, RI, 1992.  Google Scholar

[7]

L. A. Cherkas, Estimation of the number of limit cycles of autonomous systems, Diff. Eqs., 13 (1977), 529-547. Google Scholar

[8]

J. B. Li and L. J. Zhang, Bifurcations of traveling wave solutions in generalized Pochhammer-Chree equation, Chaos. Soliton. Fract., 14 (2002), 581-593. doi: 10.1016/S0960-0779(01)00248-X.  Google Scholar

[9]

Q. X. Ye and Z. Y. Li, "Introduction of Reaction Diffusion Equations," Science Press, Beijing, 1990. (in Chinese)  Google Scholar

[10]

P. C. Fife, "Mathematical Aspects of Reacting and Diffusing Systems," Springer-Verlag, Berlin-New York, 1979.  Google Scholar

[11]

D. G. Aronson and H. F. Weiberger, Multidimentional nonlinear diffusion arising in population genetics, Adv.in Math., 30 (1978), 33-76. doi: doi:10.1016/0001-8708(78)90130-5.  Google Scholar

[12]

A. Jeffery and T. Kakutani, Stability of the Burgers shock wave and the Korteweg-de Vries soliton,, Indiana Univ. Math. J., 20 (): 463.  doi: 10.1512/iumj.1970.20.20039.  Google Scholar

[13]

J. L. Bona and M. E. Schonbek, Travelling wave solutions to the Korteweg-de Vries-Burgers equation, Proc. R. Soc. Edinburgh Sect. A, 101 (1985), 207-226. doi: 10.1017/S0308210500020783.  Google Scholar

[14]

H. Grad and P. N. Hu, Unified shock profile in a plasma, Phys. Fluids, 10 (1976), 2596-2602. doi: 10.1063/1.1762081.  Google Scholar

[15]

R. S. Johnson, A nonlinear equation incorporating damping and dispersion, J. Fluid Mech., 42 (1970), 49-60. doi: 10.1017/S0022112070001064.  Google Scholar

[16]

J. Canosa and J. Gazdag, The Korteweg-de Vries-Burgers equation, J. Computational Phys., 23 (1977), 393-403. doi: 0021-9991(77)90070-5.  Google Scholar

[17]

T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. R. Soc. London Ser. A, 272 (1972), 47-78. doi: 10.1098/rsta.1972.0032.  Google Scholar

[18]

J. L. Bona and V. A. Dougalis, An initial and boundary value problem for a model equation for propagation of long waves, J. Math. Anal. Appl., 75 (1980), 502-522. doi: 10.1016/0022-247X(80)90098-0.  Google Scholar

[19]

G. H. Ganser and D. A. Drew, Nonlinear periodic waves in a two-phase flow model, SIAM J. Appl. Math., 47 (1987), 726-736. doi: 10.1137/0147050.  Google Scholar

[20]

G. H. Ganser and D. A. Drew, Nonlinear stability analysis of a uniformly fluidized bed, Int. J. Multiphase Flow, 16 (1990), 447-460. doi: 10.1016/0301-9322(90)90075-T.  Google Scholar

[21]

L. Abia, I. Christie and J. M. Sanz-Serna, Stability of schemes for the numerical treatment of an equation modelling fluidized beds, Model. Math. Anal. Numer., 23 (1989), 191-204. doi: 0674.76022.  Google Scholar

[22]

J. C. Lopez-Marcos and J. M. Sanz-Serna, A definition of stability for nonlinear problems, Numerical Treatment of Differntial Equations, 104 (1988), 216-226.  Google Scholar

[23]

M. F. Feng and P. B. Ming, Stability finite difference method and the nonlinear stability analysis of modelling fluidized beds equation, Journal of Computational Mathematics in Colleges and Universities, 9 (1997), 298-311. Google Scholar

[1]

Jorge A. Esquivel-Avila. Qualitative analysis of a nonlinear wave equation. Discrete & Continuous Dynamical Systems, 2004, 10 (3) : 787-804. doi: 10.3934/dcds.2004.10.787
[2]

Na An, Chaobao Huang, Xijun Yu. Error analysis of discontinuous Galerkin method for the time fractional KdV equation with weak singularity solution. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 321-334. doi: 10.3934/dcdsb.2019185
[3]

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
[4]

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
[5]

Zhaosheng Feng, Yu Huang. Approximate solution of the Burgers-Korteweg-de Vries equation. Communications on Pure & Applied Analysis, 2007, 6 (2) : 429-440. doi: 10.3934/cpaa.2007.6.429
[6]

Shaoyong Lai, Yong Hong Wu, Xu Yang. The global solution of an initial boundary value problem for the damped Boussinesq equation. Communications on Pure & Applied Analysis, 2004, 3 (2) : 319-328. doi: 10.3934/cpaa.2004.3.319
[7]

Dominika Pilarczyk. Asymptotic stability of singular solution to nonlinear heat equation. Discrete & Continuous Dynamical Systems, 2009, 25 (3) : 991-1001. doi: 10.3934/dcds.2009.25.991
[8]

Kamil Aida-Zade, Jamila Asadova. Numerical solution to optimal control problems of oscillatory processes. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021166
[9]

Ludovick Gagnon. Qualitative description of the particle trajectories for the N-solitons solution of the Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1489-1507. doi: 10.3934/dcds.2017061
[10]

Fabrice Planchon, John G. Stalker, A. Shadi Tahvildar-Zadeh. Dispersive estimate for the wave equation with the inverse-square potential. Discrete & Continuous Dynamical Systems, 2003, 9 (6) : 1387-1400. doi: 10.3934/dcds.2003.9.1387
[11]

Guoshan Zhang, Shiwei Wang, Yiming Wang, Wanquan Liu. LS-SVM approximate solution for affine nonlinear systems with partially unknown functions. Journal of Industrial & Management Optimization, 2014, 10 (2) : 621-636. doi: 10.3934/jimo.2014.10.621
[12]

Zhaosheng Feng, Goong Chen, Sze-Bi Hsu. A qualitative study of the damped duffing equation and applications. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1097-1112. doi: 10.3934/dcdsb.2006.6.1097
[13]

Qiusheng Qiu, Xinmin Yang. Scalarization of approximate solution for vector equilibrium problems. Journal of Industrial & Management Optimization, 2013, 9 (1) : 143-151. doi: 10.3934/jimo.2013.9.143
[14]

Lam Quoc Anh, Pham Thanh Duoc, Tran Ngoc Tam. Continuity of approximate solution maps to vector equilibrium problems. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1685-1699. doi: 10.3934/jimo.2017013
[15]

Wenxia Chen, Ping Yang, Weiwei Gao, Lixin Tian. The approximate solution for Benjamin-Bona-Mahony equation under slowly varying medium. Communications on Pure & Applied Analysis, 2018, 17 (3) : 823-848. doi: 10.3934/cpaa.2018042
[16]

Editorial Office. WITHDRAWN: Fractional diffusion equation described by the Atangana-Baleanu fractional derivative and its approximate solution. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2020173
[17]

Olivier Goubet. Approximate inertial manifolds for a weakly damped nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 1997, 3 (4) : 503-530. doi: 10.3934/dcds.1997.3.503
[18]

Kaïs Ammari, Thomas Duyckaerts, Armen Shirikyan. Local feedback stabilisation to a non-stationary solution for a damped non-linear wave equation. Mathematical Control & Related Fields, 2016, 6 (1) : 1-25. doi: 10.3934/mcrf.2016.6.1
[19]

Bassam Kojok. Global existence for a forced dispersive dissipative equation via the I-method. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1401-1419. doi: 10.3934/cpaa.2009.8.1401
[20]

T. Diogo, P. Lima, M. Rebelo. Numerical solution of a nonlinear Abel type Volterra integral equation. Communications on Pure & Applied Analysis, 2006, 5 (2) : 277-288. doi: 10.3934/cpaa.2006.5.277

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (50)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences