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Travelling wave solutions of a free boundary problem for a two-species competitive model
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 |
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. |
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. |
[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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