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Global dynamics of three species omnivory models with Lotka-Volterra interaction
Bounded traveling wave solutions for MKdV-Burgers equation with the negative dispersive coefficient
1. | School of Science, University of Shanghai for Science and Technology, Shanghai 200093, China |
2. | University of Shanghai for Science and Technology, Shanghai 200093, China, China, China, China |
References:
[1] |
G. P. Agrawal, Nonlinear Fibre Optics, Academic Press, Boston, 1989. |
[2] |
D. G. Aronson and H. F. Weibererger, Multidimentional nonlinear diffusion arising in population genetics, Adv. Math., 30 (1978), 33-76.
doi: 10.1016/0001-8708(78)90130-5. |
[3] |
R. Beals, P. Deift and C. Tomei, Direct and Inverse Scattering on the Line, Mathematical Surveys and Monographs, vol. 28, Amer. Math. Soc. Providence, RI, 1988.
doi: 10.1090/surv/028. |
[4] |
A. Bekir, On travelling wave solutions to combined KdV-mKdV equation and modified Burgers- KdV equation, Commun. Nonlinear. Sci. Numer. Simulat., 14 (2009), 1038-1042.
doi: 10.1016/j.cnsns.2008.03.014. |
[5] |
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. |
[6] |
D. J. Benney, Long waves on liquid films, J. Math. Phys., 45 (1966), 150-155.
doi: 10.1002/sapm1966451150. |
[7] |
R. F. Bikbaev, Shock waves in the modified Burgers-Korteweg-de-Vries equation, J. Nonlinear Sci., 5 (1995), 1-10.
doi: 10.1007/BF01869099. |
[8] |
J. L. Bona and V. A. Dougalia, An intial-and boundary-value problem for a model equation for propagation of long waves, J. Math. Anal. Appl., 75 (1980), 503-522.
doi: 10.1016/0022-247X(80)90098-0. |
[9] |
J. L. Bona and M. E. Schonbek, Travelling wave olutions to the Korteweg-de Vries-Burgers equation, Proc. R. Soc. Edin., 101 (1985), 207-226.
doi: 10.1017/S0308210500020783. |
[10] |
E. F. EL-Shamy, Dust-ion-acoustic solitary waves in a hot magnetized dusty plasma with charge fluctuations, Chaos Soliton. Fract., 25 (2005), 665-674.
doi: 10.1016/j.chaos.2004.11.047. |
[11] |
Z. S. Feng, On travelling wave solutions to modified Burgers-Korteweg-de-Vries equation, Phys. Lett. A, 318 (2003), 522-525.
doi: 10.1016/j.physleta.2003.09.057. |
[12] |
P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics, 28, Springer-Verlag, New York, 1979. |
[13] |
J. Ginibre, Y. Tsutsumi and G. Velo, Uniqueness of solutions for the generalized Korteweg-de Vries equation, Siam J. Math. Anal., 20 (1989), 1388-1425.
doi: 10.1137/0520091. |
[14] |
H. Grad and P. N. Hu, Unified shock profile in a plasma, Phys. Fluids., 10 (1967), p2596.
doi: 10.1063/1.1762081. |
[15] |
P. N. Hu, Collisional theory of shock and nonlinear waves in a plasma, Phys. Fluids., 15 (1972), 854-864.
doi: 10.1063/1.1693994. |
[16] |
D. Jacobs, B. Mckinney and M. Shearer, Travelling wave solutions of the modified Korteweg-deVries-Burgers equation, J. Differ. Equ., 116 (1995), 448-467.
doi: 10.1006/jdeq.1995.1043. |
[17] |
R. S. Johnson, Shallow water waves on a viscous fluid-the undular bore, Phys. Fluids., 15 (1972), 1693-1699.
doi: 10.1063/1.1693764. |
[18] |
R. S. Johnson, A nonlinear incorporating damping and dispersion, J. Fluid. Mech., 42 (1970), 49-60.
doi: 10.1017/S0022112070001064. |
[19] |
Y. Kametaka, Korteweg-de Vries equation IV. Simplest generalization, Proc. Japan Acad., 45 (1969), 661-665.
doi: 10.3792/pja/1195520615. |
[20] |
T. Kato, On the Korteweg-de Vries equation, Manuscripta Math., 28 (1979), 89-99.
doi: 10.1007/BF01647967. |
[21] |
T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Studies in applied mathematics, (V. Guillemin, ed.), Adv. Math. Suppl., Studies, Academic Press, New York, 8 (1983), 93-128. |
[22] |
S. N. Kruzhkov and A. V. Faminskii, Generalized solutions of the Cauchy problem for the Korteweg-de Vries equation, (Russian) Mat. Sb. (N.S.), 120 (1983), 396-425.
doi: 10.1070/SM1984v048n02ABEH002682. |
[23] |
H. B. Li and P. H. Huang, Simulation of the MKdV equation with lattice Boltzmann method, Acta Physica Sinica., 50 (2001), 837-840. (in Chinese) |
[24] |
R. M. Miura, Korteweg-de Vries Equation and Generalizations. I. A Remarkable Explicit Nonlinear Transformation, J. Math. Phys., 9 (1968), 1202-1204.
doi: 10.1063/1.1664700. |
[25] |
V. Nemytskii and V. Stepanov, Qualitative Theory of Differential Equations, Princeton University Press, Princeton, N.J. 1960.
doi: 10.1515/9781400875955. |
[26] |
S. P. Novikov, S. V. Manakov, L. P. Pitaevskii and V. E. Zakharov, Theory of Solitons, the Inverse Scattering Methods, Nauka, Moskva, 1980. |
[27] |
M. Ohmiya, On the generalized soliton solutions of the modified Korteweg-de Vries equation, Osaka J. Math., 11 (1974), 61-71. |
[28] |
Y. R. Shi and P. Guo, Expansion method for modified Jacobi elliptic function and its application, Acta Physica Sinica., 53 (2004), 3265-3269. (in Chinese) |
[29] |
S. Tanaka, Non-linear Schrödinger equation and modified Korteweg-de Vries equation; construction of solutions in terms of scattering data, Publ. Res. Inst. Math. Soc., 10 (1975), 329-357.
doi: 10.2977/prims/1195191998. |
[30] |
S. Tanaka, Some remarks on the modified Korteweg-de Vries equation,, Publ. Res. Inst. Math. Sei., 8 (): 429.
doi: 10.2977/prims/1195192956. |
[31] |
J. S. Tang, Z. Y. Liu and X. P. Li, The quasi wavelet solution of MKdV equation, Acta Physica Sinica., 52 (2003), 522-525. (in Chinese) |
[32] |
M. Tsutsumi, On global solutions of the generalized Korteweg-de Vries equation, Publ. Res.Inst. Math. Soc., 7 (1972), 329-344.
doi: 10.2977/prims/1195193545. |
[33] |
M. Wadati, The Modified Korteweg-de Vries Equation, J. Phys. Soc. Japan., 34 (1973), 1289-1296.
doi: 10.1143/JPSJ.34.1289. |
[34] |
L. V. Wijngaarden, On the motion of gas bubbles in a perfect fluid, Arch. Mech., 34 (1982), 343-349. |
[35] |
Q. Ye and Z. Li, Introduction of Reaction-Diffusion Equations, Science Press, Beijing, 1990. (in Chinese) |
[36] |
Z. F. Zhang, T. R. Ding and W. S. Huang, Qualitative Theory of Differential Equations, American Mathematical Society, Providence, RI, 1992. |
[37] |
W. G. Zhang, Q. S. Chang and B. G. Jiang, Explicit exact solitary-wave solutions for compound KdV-type and compound KdV Burgers-type equations with nonlinear term of any order, Chaos Soliton. Fract., 13 (2002), 311-319.
doi: 10.1016/S0960-0779(00)00272-1. |
[38] |
W. G. Zhang, J. Xu, X. Li and Y. Zhao, Approximate damped oscillatory solutions for MKdV-Burgers equation and their error estimates, Journal of University of Shanghai for Science and Technology, 34 (2012), 409-418. (in Chinese)
doi: 10.13255/j.cnki.jusst.2012.05.001. |
[39] |
S. Zhao and B. Xu, The inverse scattering solutions of MKdV equation, Appl. Math. J. Chinese. Univ., 4 (1989), 398-402. (in Chinese)
doi: 10.13299/j.cnki.amjcu.000229. |
show all references
References:
[1] |
G. P. Agrawal, Nonlinear Fibre Optics, Academic Press, Boston, 1989. |
[2] |
D. G. Aronson and H. F. Weibererger, Multidimentional nonlinear diffusion arising in population genetics, Adv. Math., 30 (1978), 33-76.
doi: 10.1016/0001-8708(78)90130-5. |
[3] |
R. Beals, P. Deift and C. Tomei, Direct and Inverse Scattering on the Line, Mathematical Surveys and Monographs, vol. 28, Amer. Math. Soc. Providence, RI, 1988.
doi: 10.1090/surv/028. |
[4] |
A. Bekir, On travelling wave solutions to combined KdV-mKdV equation and modified Burgers- KdV equation, Commun. Nonlinear. Sci. Numer. Simulat., 14 (2009), 1038-1042.
doi: 10.1016/j.cnsns.2008.03.014. |
[5] |
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. |
[6] |
D. J. Benney, Long waves on liquid films, J. Math. Phys., 45 (1966), 150-155.
doi: 10.1002/sapm1966451150. |
[7] |
R. F. Bikbaev, Shock waves in the modified Burgers-Korteweg-de-Vries equation, J. Nonlinear Sci., 5 (1995), 1-10.
doi: 10.1007/BF01869099. |
[8] |
J. L. Bona and V. A. Dougalia, An intial-and boundary-value problem for a model equation for propagation of long waves, J. Math. Anal. Appl., 75 (1980), 503-522.
doi: 10.1016/0022-247X(80)90098-0. |
[9] |
J. L. Bona and M. E. Schonbek, Travelling wave olutions to the Korteweg-de Vries-Burgers equation, Proc. R. Soc. Edin., 101 (1985), 207-226.
doi: 10.1017/S0308210500020783. |
[10] |
E. F. EL-Shamy, Dust-ion-acoustic solitary waves in a hot magnetized dusty plasma with charge fluctuations, Chaos Soliton. Fract., 25 (2005), 665-674.
doi: 10.1016/j.chaos.2004.11.047. |
[11] |
Z. S. Feng, On travelling wave solutions to modified Burgers-Korteweg-de-Vries equation, Phys. Lett. A, 318 (2003), 522-525.
doi: 10.1016/j.physleta.2003.09.057. |
[12] |
P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics, 28, Springer-Verlag, New York, 1979. |
[13] |
J. Ginibre, Y. Tsutsumi and G. Velo, Uniqueness of solutions for the generalized Korteweg-de Vries equation, Siam J. Math. Anal., 20 (1989), 1388-1425.
doi: 10.1137/0520091. |
[14] |
H. Grad and P. N. Hu, Unified shock profile in a plasma, Phys. Fluids., 10 (1967), p2596.
doi: 10.1063/1.1762081. |
[15] |
P. N. Hu, Collisional theory of shock and nonlinear waves in a plasma, Phys. Fluids., 15 (1972), 854-864.
doi: 10.1063/1.1693994. |
[16] |
D. Jacobs, B. Mckinney and M. Shearer, Travelling wave solutions of the modified Korteweg-deVries-Burgers equation, J. Differ. Equ., 116 (1995), 448-467.
doi: 10.1006/jdeq.1995.1043. |
[17] |
R. S. Johnson, Shallow water waves on a viscous fluid-the undular bore, Phys. Fluids., 15 (1972), 1693-1699.
doi: 10.1063/1.1693764. |
[18] |
R. S. Johnson, A nonlinear incorporating damping and dispersion, J. Fluid. Mech., 42 (1970), 49-60.
doi: 10.1017/S0022112070001064. |
[19] |
Y. Kametaka, Korteweg-de Vries equation IV. Simplest generalization, Proc. Japan Acad., 45 (1969), 661-665.
doi: 10.3792/pja/1195520615. |
[20] |
T. Kato, On the Korteweg-de Vries equation, Manuscripta Math., 28 (1979), 89-99.
doi: 10.1007/BF01647967. |
[21] |
T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Studies in applied mathematics, (V. Guillemin, ed.), Adv. Math. Suppl., Studies, Academic Press, New York, 8 (1983), 93-128. |
[22] |
S. N. Kruzhkov and A. V. Faminskii, Generalized solutions of the Cauchy problem for the Korteweg-de Vries equation, (Russian) Mat. Sb. (N.S.), 120 (1983), 396-425.
doi: 10.1070/SM1984v048n02ABEH002682. |
[23] |
H. B. Li and P. H. Huang, Simulation of the MKdV equation with lattice Boltzmann method, Acta Physica Sinica., 50 (2001), 837-840. (in Chinese) |
[24] |
R. M. Miura, Korteweg-de Vries Equation and Generalizations. I. A Remarkable Explicit Nonlinear Transformation, J. Math. Phys., 9 (1968), 1202-1204.
doi: 10.1063/1.1664700. |
[25] |
V. Nemytskii and V. Stepanov, Qualitative Theory of Differential Equations, Princeton University Press, Princeton, N.J. 1960.
doi: 10.1515/9781400875955. |
[26] |
S. P. Novikov, S. V. Manakov, L. P. Pitaevskii and V. E. Zakharov, Theory of Solitons, the Inverse Scattering Methods, Nauka, Moskva, 1980. |
[27] |
M. Ohmiya, On the generalized soliton solutions of the modified Korteweg-de Vries equation, Osaka J. Math., 11 (1974), 61-71. |
[28] |
Y. R. Shi and P. Guo, Expansion method for modified Jacobi elliptic function and its application, Acta Physica Sinica., 53 (2004), 3265-3269. (in Chinese) |
[29] |
S. Tanaka, Non-linear Schrödinger equation and modified Korteweg-de Vries equation; construction of solutions in terms of scattering data, Publ. Res. Inst. Math. Soc., 10 (1975), 329-357.
doi: 10.2977/prims/1195191998. |
[30] |
S. Tanaka, Some remarks on the modified Korteweg-de Vries equation,, Publ. Res. Inst. Math. Sei., 8 (): 429.
doi: 10.2977/prims/1195192956. |
[31] |
J. S. Tang, Z. Y. Liu and X. P. Li, The quasi wavelet solution of MKdV equation, Acta Physica Sinica., 52 (2003), 522-525. (in Chinese) |
[32] |
M. Tsutsumi, On global solutions of the generalized Korteweg-de Vries equation, Publ. Res.Inst. Math. Soc., 7 (1972), 329-344.
doi: 10.2977/prims/1195193545. |
[33] |
M. Wadati, The Modified Korteweg-de Vries Equation, J. Phys. Soc. Japan., 34 (1973), 1289-1296.
doi: 10.1143/JPSJ.34.1289. |
[34] |
L. V. Wijngaarden, On the motion of gas bubbles in a perfect fluid, Arch. Mech., 34 (1982), 343-349. |
[35] |
Q. Ye and Z. Li, Introduction of Reaction-Diffusion Equations, Science Press, Beijing, 1990. (in Chinese) |
[36] |
Z. F. Zhang, T. R. Ding and W. S. Huang, Qualitative Theory of Differential Equations, American Mathematical Society, Providence, RI, 1992. |
[37] |
W. G. Zhang, Q. S. Chang and B. G. Jiang, Explicit exact solitary-wave solutions for compound KdV-type and compound KdV Burgers-type equations with nonlinear term of any order, Chaos Soliton. Fract., 13 (2002), 311-319.
doi: 10.1016/S0960-0779(00)00272-1. |
[38] |
W. G. Zhang, J. Xu, X. Li and Y. Zhao, Approximate damped oscillatory solutions for MKdV-Burgers equation and their error estimates, Journal of University of Shanghai for Science and Technology, 34 (2012), 409-418. (in Chinese)
doi: 10.13255/j.cnki.jusst.2012.05.001. |
[39] |
S. Zhao and B. Xu, The inverse scattering solutions of MKdV equation, Appl. Math. J. Chinese. Univ., 4 (1989), 398-402. (in Chinese)
doi: 10.13299/j.cnki.amjcu.000229. |
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