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 & 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.  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.  doi: 10.1143/JPSJ.29.1068.  Google Scholar

[3]

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

[4]

N. Isidore, Exact solutions of a nonlinear dispersive-dissipative equation,, J. Phys. A, 29 (1996), 3679.  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, (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, (1992).   Google Scholar

[7]

L. A. Cherkas, Estimation of the number of limit cycles of autonomous systems,, Diff. Eqs., 13 (1977), 529.   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.  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, (1990).   Google Scholar

[10]

P. C. Fife, "Mathematical Aspects of Reacting and Diffusing Systems,", Springer-Verlag, (1979).   Google Scholar

[11]

D. G. Aronson and H. F. Weiberger, Multidimentional nonlinear diffusion arising in population genetics,, Adv.in Math., 30 (1978), 33.  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.  doi: 10.1017/S0308210500020783.  Google Scholar

[14]

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

[15]

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

[16]

J. Canosa and J. Gazdag, The Korteweg-de Vries-Burgers equation,, J. Computational Phys., 23 (1977), 393.  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.  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.  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.  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.  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.  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.   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.   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.  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.  doi: 10.1143/JPSJ.29.1068.  Google Scholar

[3]

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

[4]

N. Isidore, Exact solutions of a nonlinear dispersive-dissipative equation,, J. Phys. A, 29 (1996), 3679.  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, (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, (1992).   Google Scholar

[7]

L. A. Cherkas, Estimation of the number of limit cycles of autonomous systems,, Diff. Eqs., 13 (1977), 529.   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.  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, (1990).   Google Scholar

[10]

P. C. Fife, "Mathematical Aspects of Reacting and Diffusing Systems,", Springer-Verlag, (1979).   Google Scholar

[11]

D. G. Aronson and H. F. Weiberger, Multidimentional nonlinear diffusion arising in population genetics,, Adv.in Math., 30 (1978), 33.  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.  doi: 10.1017/S0308210500020783.  Google Scholar

[14]

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

[15]

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

[16]

J. Canosa and J. Gazdag, The Korteweg-de Vries-Burgers equation,, J. Computational Phys., 23 (1977), 393.  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.  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.  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.  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.  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.  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.   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.   Google Scholar

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