# American Institute of Mathematical Sciences

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

 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.
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 and 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. [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. [3] M. Malfliet, "The Tanh Method in Nonlinear Wave Theory, Habilitation Thesis," University of Antwerp, Antwerp, 1994. [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. [5] V. V. Nemytskii and V. V. Stepanov, "Qualitative Theory of Differential Equations," Princeton University Press, Princeton, 1989. [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. [7] L. A. Cherkas, Estimation of the number of limit cycles of autonomous systems, Diff. Eqs., 13 (1977), 529-547. [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. [9] Q. X. Ye and Z. Y. Li, "Introduction of Reaction Diffusion Equations," Science Press, Beijing, 1990. (in Chinese) [10] P. C. Fife, "Mathematical Aspects of Reacting and Diffusing Systems," Springer-Verlag, Berlin-New York, 1979. [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. [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. [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. [14] H. Grad and P. N. Hu, Unified shock profile in a plasma, Phys. Fluids, 10 (1976), 2596-2602. doi: 10.1063/1.1762081. [15] R. S. Johnson, A nonlinear equation incorporating damping and dispersion, J. Fluid Mech., 42 (1970), 49-60. doi: 10.1017/S0022112070001064. [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. [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. [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. [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. [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. [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. [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. [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.

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##### 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. [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. [3] M. Malfliet, "The Tanh Method in Nonlinear Wave Theory, Habilitation Thesis," University of Antwerp, Antwerp, 1994. [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. [5] V. V. Nemytskii and V. V. Stepanov, "Qualitative Theory of Differential Equations," Princeton University Press, Princeton, 1989. [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. [7] L. A. Cherkas, Estimation of the number of limit cycles of autonomous systems, Diff. Eqs., 13 (1977), 529-547. [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. [9] Q. X. Ye and Z. Y. Li, "Introduction of Reaction Diffusion Equations," Science Press, Beijing, 1990. (in Chinese) [10] P. C. Fife, "Mathematical Aspects of Reacting and Diffusing Systems," Springer-Verlag, Berlin-New York, 1979. [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. [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. [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. [14] H. Grad and P. N. Hu, Unified shock profile in a plasma, Phys. Fluids, 10 (1976), 2596-2602. doi: 10.1063/1.1762081. [15] R. S. Johnson, A nonlinear equation incorporating damping and dispersion, J. Fluid Mech., 42 (1970), 49-60. doi: 10.1017/S0022112070001064. [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. [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. [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. [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. [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. [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. [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. [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.
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