American Institute of Mathematical Sciences

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

Bounded traveling wave solutions for MKdV-Burgers equation with the negative dispersive coefficient

Weiguo Zhang 1, , Yujiao Sun 2, , Zhengming Li 2, , Shengbing Pei 2, and  Xiang Li 2,

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.
Keywords: damped oscillatory solution., qualitative analysis, traveling wave solutions, MKdV-Burgers equation.
Mathematics Subject Classification: Primary: 35Q51, 37C29; Secondary: 37C2.
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 & Continuous Dynamical Systems - B, 2016, 21 (8) : 2883-2903. doi: 10.3934/dcdsb.2016078
References:
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G. P. Agrawal, Nonlinear Fibre Optics,, Academic Press, (1989). Google Scholar

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D. G. Aronson and H. F. Weibererger, Multidimentional nonlinear diffusion arising in population genetics,, Adv. Math., 30 (1978), 33. doi: 10.1016/0001-8708(78)90130-5. Google Scholar

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A. Bekir, On travelling wave solutions to combined KdV-mKdV equation and modified Burgers- KdV equation,, Commun. Nonlinear. Sci. Numer. Simulat., 14 (2009), 1038. doi: 10.1016/j.cnsns.2008.03.014. Google Scholar

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D. J. Benney, Long waves on liquid films,, J. Math. Phys., 45 (1966), 150. doi: 10.1002/sapm1966451150. Google Scholar

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R. F. Bikbaev, Shock waves in the modified Burgers-Korteweg-de-Vries equation,, J. Nonlinear Sci., 5 (1995), 1. doi: 10.1007/BF01869099. Google Scholar

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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. doi: 10.1016/0022-247X(80)90098-0. Google Scholar

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J. L. Bona and M. E. Schonbek, Travelling wave olutions to the Korteweg-de Vries-Burgers equation,, Proc. R. Soc. Edin., 101 (1985), 207. doi: 10.1017/S0308210500020783. Google Scholar

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J. Ginibre, Y. Tsutsumi and G. Velo, Uniqueness of solutions for the generalized Korteweg-de Vries equation,, Siam J. Math. Anal., 20 (1989), 1388. doi: 10.1137/0520091. Google Scholar

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H. Grad and P. N. Hu, Unified shock profile in a plasma,, Phys. Fluids., 10 (1967). doi: 10.1063/1.1762081. Google Scholar

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P. N. Hu, Collisional theory of shock and nonlinear waves in a plasma,, Phys. Fluids., 15 (1972), 854. doi: 10.1063/1.1693994. Google Scholar

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D. Jacobs, B. Mckinney and M. Shearer, Travelling wave solutions of the modified Korteweg-deVries-Burgers equation,, J. Differ. Equ., 116 (1995), 448. doi: 10.1006/jdeq.1995.1043. Google Scholar

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R. S. Johnson, Shallow water waves on a viscous fluid-the undular bore,, Phys. Fluids., 15 (1972), 1693. doi: 10.1063/1.1693764. Google Scholar

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

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Y. Kametaka, Korteweg-de Vries equation IV. Simplest generalization,, Proc. Japan Acad., 45 (1969), 661. doi: 10.3792/pja/1195520615. Google Scholar

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T. Kato, On the Korteweg-de Vries equation,, Manuscripta Math., 28 (1979), 89. doi: 10.1007/BF01647967. Google Scholar

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T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Studies in applied mathematics, (V. Guillemin, ed.),, Adv. Math. Suppl., 8 (1983), 93. Google Scholar

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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. doi: 10.1070/SM1984v048n02ABEH002682. Google Scholar

[23]

H. B. Li and P. H. Huang, Simulation of the MKdV equation with lattice Boltzmann method,, Acta Physica Sinica., 50 (2001), 837. Google Scholar

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R. M. Miura, Korteweg-de Vries Equation and Generalizations. I. A Remarkable Explicit Nonlinear Transformation,, J. Math. Phys., 9 (1968), 1202. doi: 10.1063/1.1664700. Google Scholar

[25]

V. Nemytskii and V. Stepanov, Qualitative Theory of Differential Equations,, Princeton University Press, (1960). doi: 10.1515/9781400875955. Google Scholar

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S. P. Novikov, S. V. Manakov, L. P. Pitaevskii and V. E. Zakharov, Theory of Solitons, the Inverse Scattering Methods,, Nauka, (1980). Google Scholar

[27]

M. Ohmiya, On the generalized soliton solutions of the modified Korteweg-de Vries equation,, Osaka J. Math., 11 (1974), 61. Google Scholar

[28]

Y. R. Shi and P. Guo, Expansion method for modified Jacobi elliptic function and its application,, Acta Physica Sinica., 53 (2004), 3265. Google Scholar

[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. doi: 10.2977/prims/1195191998. Google Scholar

[30]

S. Tanaka, Some remarks on the modified Korteweg-de Vries equation,, Publ. Res. Inst. Math. Sei., 8 (): 429. doi: 10.2977/prims/1195192956. Google Scholar

[31]

J. S. Tang, Z. Y. Liu and X. P. Li, The quasi wavelet solution of MKdV equation,, Acta Physica Sinica., 52 (2003), 522. Google Scholar

[32]

M. Tsutsumi, On global solutions of the generalized Korteweg-de Vries equation,, Publ. Res.Inst. Math. Soc., 7 (1972), 329. doi: 10.2977/prims/1195193545. Google Scholar

[33]

M. Wadati, The Modified Korteweg-de Vries Equation,, J. Phys. Soc. Japan., 34 (1973), 1289. doi: 10.1143/JPSJ.34.1289. Google Scholar

[34]

L. V. Wijngaarden, On the motion of gas bubbles in a perfect fluid,, Arch. Mech., 34 (1982), 343. Google Scholar

[35]

Q. Ye and Z. Li, Introduction of Reaction-Diffusion Equations,, Science Press, (1990). Google Scholar

[36]

Z. F. Zhang, T. R. Ding and W. S. Huang, Qualitative Theory of Differential Equations,, American Mathematical Society, (1992). Google Scholar

[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. doi: 10.1016/S0960-0779(00)00272-1. Google Scholar

[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. doi: 10.13255/j.cnki.jusst.2012.05.001. Google Scholar

[39]

S. Zhao and B. Xu, The inverse scattering solutions of MKdV equation,, Appl. Math. J. Chinese. Univ., 4 (1989), 398. doi: 10.13299/j.cnki.amjcu.000229. Google Scholar

show all references

References:
[1]

G. P. Agrawal, Nonlinear Fibre Optics,, Academic Press, (1989). Google Scholar

[2]

D. G. Aronson and H. F. Weibererger, Multidimentional nonlinear diffusion arising in population genetics,, Adv. Math., 30 (1978), 33. doi: 10.1016/0001-8708(78)90130-5. Google Scholar

[3]

R. Beals, P. Deift and C. Tomei, Direct and Inverse Scattering on the Line, Mathematical Surveys and Monographs,, vol. 28, (1988). doi: 10.1090/surv/028. Google Scholar

[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. doi: 10.1016/j.cnsns.2008.03.014. Google Scholar

[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. doi: 10.1098/rsta.1972.0032. Google Scholar

[6]

D. J. Benney, Long waves on liquid films,, J. Math. Phys., 45 (1966), 150. doi: 10.1002/sapm1966451150. Google Scholar

[7]

R. F. Bikbaev, Shock waves in the modified Burgers-Korteweg-de-Vries equation,, J. Nonlinear Sci., 5 (1995), 1. doi: 10.1007/BF01869099. Google Scholar

[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. doi: 10.1016/0022-247X(80)90098-0. Google Scholar

[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. doi: 10.1017/S0308210500020783. Google Scholar

[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. doi: 10.1016/j.chaos.2004.11.047. Google Scholar

[11]

Z. S. Feng, On travelling wave solutions to modified Burgers-Korteweg-de-Vries equation,, Phys. Lett. A, 318 (2003), 522. doi: 10.1016/j.physleta.2003.09.057. Google Scholar

[12]

P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics, 28,, Springer-Verlag, (1979). Google Scholar

[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. doi: 10.1137/0520091. Google Scholar

[14]

H. Grad and P. N. Hu, Unified shock profile in a plasma,, Phys. Fluids., 10 (1967). doi: 10.1063/1.1762081. Google Scholar

[15]

P. N. Hu, Collisional theory of shock and nonlinear waves in a plasma,, Phys. Fluids., 15 (1972), 854. doi: 10.1063/1.1693994. Google Scholar

[16]

D. Jacobs, B. Mckinney and M. Shearer, Travelling wave solutions of the modified Korteweg-deVries-Burgers equation,, J. Differ. Equ., 116 (1995), 448. doi: 10.1006/jdeq.1995.1043. Google Scholar

[17]

R. S. Johnson, Shallow water waves on a viscous fluid-the undular bore,, Phys. Fluids., 15 (1972), 1693. doi: 10.1063/1.1693764. Google Scholar

[18]

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

[19]

Y. Kametaka, Korteweg-de Vries equation IV. Simplest generalization,, Proc. Japan Acad., 45 (1969), 661. doi: 10.3792/pja/1195520615. Google Scholar

[20]

T. Kato, On the Korteweg-de Vries equation,, Manuscripta Math., 28 (1979), 89. doi: 10.1007/BF01647967. Google Scholar

[21]

T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Studies in applied mathematics, (V. Guillemin, ed.),, Adv. Math. Suppl., 8 (1983), 93. Google Scholar

[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. doi: 10.1070/SM1984v048n02ABEH002682. Google Scholar

[23]

H. B. Li and P. H. Huang, Simulation of the MKdV equation with lattice Boltzmann method,, Acta Physica Sinica., 50 (2001), 837. Google Scholar

[24]

R. M. Miura, Korteweg-de Vries Equation and Generalizations. I. A Remarkable Explicit Nonlinear Transformation,, J. Math. Phys., 9 (1968), 1202. doi: 10.1063/1.1664700. Google Scholar

[25]

V. Nemytskii and V. Stepanov, Qualitative Theory of Differential Equations,, Princeton University Press, (1960). doi: 10.1515/9781400875955. Google Scholar

[26]

S. P. Novikov, S. V. Manakov, L. P. Pitaevskii and V. E. Zakharov, Theory of Solitons, the Inverse Scattering Methods,, Nauka, (1980). Google Scholar

[27]

M. Ohmiya, On the generalized soliton solutions of the modified Korteweg-de Vries equation,, Osaka J. Math., 11 (1974), 61. Google Scholar

[28]

Y. R. Shi and P. Guo, Expansion method for modified Jacobi elliptic function and its application,, Acta Physica Sinica., 53 (2004), 3265. Google Scholar

[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. doi: 10.2977/prims/1195191998. Google Scholar

[30]

S. Tanaka, Some remarks on the modified Korteweg-de Vries equation,, Publ. Res. Inst. Math. Sei., 8 (): 429. doi: 10.2977/prims/1195192956. Google Scholar

[31]

J. S. Tang, Z. Y. Liu and X. P. Li, The quasi wavelet solution of MKdV equation,, Acta Physica Sinica., 52 (2003), 522. Google Scholar

[32]

M. Tsutsumi, On global solutions of the generalized Korteweg-de Vries equation,, Publ. Res.Inst. Math. Soc., 7 (1972), 329. doi: 10.2977/prims/1195193545. Google Scholar

[33]

M. Wadati, The Modified Korteweg-de Vries Equation,, J. Phys. Soc. Japan., 34 (1973), 1289. doi: 10.1143/JPSJ.34.1289. Google Scholar

[34]

L. V. Wijngaarden, On the motion of gas bubbles in a perfect fluid,, Arch. Mech., 34 (1982), 343. Google Scholar

[35]

Q. Ye and Z. Li, Introduction of Reaction-Diffusion Equations,, Science Press, (1990). Google Scholar

[36]

Z. F. Zhang, T. R. Ding and W. S. Huang, Qualitative Theory of Differential Equations,, American Mathematical Society, (1992). Google Scholar

[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. doi: 10.1016/S0960-0779(00)00272-1. Google Scholar

[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. doi: 10.13255/j.cnki.jusst.2012.05.001. Google Scholar

[39]

S. Zhao and B. Xu, The inverse scattering solutions of MKdV equation,, Appl. Math. J. Chinese. Univ., 4 (1989), 398. doi: 10.13299/j.cnki.amjcu.000229. Google Scholar

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