October  2016, 21(8): 2883-2903. doi: 10.3934/dcdsb.2016078

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

Received  August 2013 Revised  January 2016 Published  September 2016

This paper studies the bounded traveling wave solutions of MKdV-Burgers equation with the negative dispersive coefficient by the theory of planar dynamical systems, undetermined coefficients method. The global phase portraits under the different parameter conditions, as well as the existent number and conditions of the bounded traveling wave solutions are obtained for the dynamical system corresponding to the traveling wave solutions of MKdV-Burgers equation. The relation is investigated between the profiles of the bounded traveling wave solutions and dissipative coefficient. And a critical value characterizing the scale of dissipative effect, is given, which is different from the one proposed by R.F. Bikbaev in his article. Focusing on the open issue proposed by R.F. Bikbaev, based on the bell and kink profile solitary wave solutions of MKdV-Burgers equation we presented, approximate damped oscillatory solutions of MKdV-Burgers equation are obtained according to the evolution relation of orbits corresponding to the approximate damped oscillatory solutions in the global phase portraits.
Citation: Weiguo Zhang, Yujiao Sun, Zhengming Li, Shengbing Pei, Xiang Li. Bounded traveling wave solutions for MKdV-Burgers equation with the negative dispersive coefficient. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2883-2903. doi: 10.3934/dcdsb.2016078
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 (1972/73), 429-437. 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 (1972/73), 429-437. 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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