Dynamical Stability of Traveling Wave Solutions to KdV-Burgers Equation

doi:10.19596/j.cnki.1001-246x.7617

CHINESE JOURNAL OF COMPUTATIONAL PHYSICS ›› 2018, Vol. 35 ›› Issue (2): 178-186.DOI: 10.19596/j.cnki.1001-246x.7617

Previous Articles Next Articles

Dynamical Stability of Traveling Wave Solutions to KdV-Burgers Equation

SHI Yuren¹, FENG Wenxing¹, XI Zhonghong^1,2, ZONG Jin^1,2, SONG Zongbin¹, PANG Jungang¹

1. College of Physics and Electronic Engineering, Northwest Normal University, Lanzhou 730070, China;
2. College of Physics and Hydropower Engineering, Gansu Normal University For Nationalities, Hezuo 747000, China

Received:2017-01-03 Revised:2017-05-03 Online:2018-03-25 Published:2018-03-25

Abstract

Abstract: We made linearization stability analysis on traveling wave solutions of KdV-Burgers equation. Numerical results indicate that traveling waves are dynamically stable for positive-dissipation case, while they are dynamically unstable for negative-dissipation case. Then we presented a finite difference scheme, which is conditionally stable, for long-time evolution of perturbed traveling waves. Numerical results also show that traveling waves are dynamically stable as positive-dissipation is held. Our results modify and improve conclusions given in relative literatures.

Key words: KdV-Burgers equation, traveling wave solution, dynamical stability

CLC Number:

O411.1

SHI Yuren, FENG Wenxing, XI Zhonghong, ZONG Jin, SONG Zongbin, PANG Jungang. Dynamical Stability of Traveling Wave Solutions to KdV-Burgers Equation[J]. CHINESE JOURNAL OF COMPUTATIONAL PHYSICS, 2018, 35(2): 178-186.

References

[1] QIAN Z W. Nonlinear acoustic waves[M]. Beijing:Science Press, 1992:110.
[2] JOHNSON R S. Nonlinear waves in fluid-filled elastic tubes and related problems[D]. London:University of London, 1969.
[3] EMAD K E, ABEER A M, ASHRAF M T, et al. Space-time fractional KdV-Burgers equation for dust acoustic shock waves in dusty plasma with non-thermal ions[J]. Chinese Phys B, 2014, 23(7):070505.
[4] WANG D Y, WU D J, HUANG G L. Solitary waves in space plasma[M]. Shanghai:Shanghai Scientific and Technological Education Publishing House, 2000:75-84.
[5] LIU S D, LIU S K. KdV-Burgers equation modeling of turbulence[J]. Science in China(A), 1992, 35(5):66-76.
[6] LIU S D, LIU S K. KdV-Burgers equation as a model of turbulence[J]. Science in China(A), 1991, 9:938-946.
[7] GUAN K Y, GAO G. Qualitative analysis on the traveling wave solutions for the mixed Burgers-KdV type equation[J]. Science in China(A), 1987, 1:64-73.
[8] LV K P, SHI Y R, DUAN W S, et al. The solitary wave solutions to KdV-Burgers equation[J]. Acta Phys Sin, 2001, 50(11):2074-2076.
[9] ZHANG G X, LI Z B, DUAN Y S. Exact solitary wave solutions to nonlinear wave equations[J]. Science in China (A), 2000, 30(12):1103-1108.
[10] WANG L X, ZONG J, WANG X L, et al. Solitary waves with double kinks of mBBM equation and their dynamical stabilities[J]. Chinese Journal of Computational Physics, 2016, 33(2):212-220.
[11] SHI Y R,ZHOU Z G,ZHANG J,et al. Solitary wave solutions of modified coupled KdV equation and their stability[J]. Chinese Journal of Computational Physics,2012,29(2):250-256.
[12] SHI Y R, YANG H J. Application of the homotopy analysis method to solving dissipative system[J]. Acta Phys Sin, 2010, 59(1):67-74.
[13] LIAO S J, CHEN C, XU H.Beyond perturbation:Introduction to the homotopy analysis method[M]. Beijing:Science Press, 2006.
[14] LIAO S J. Homotopy analysis method in nonlinear differential equations[M]. Beijing:Springer & Higher Education Press, 2012.
[15] LI Y S. Soliton and integrable system[M]. Shanghai:Shanghai Scientific and Technological Education Publishing House, 1999.
[16] ZHU Z N. Stability of solitary wave for the KdV type equation and traveling wave for the KdV-Burgers type equation[J]. Acta Phys Sin, 1996, 45(7):1087-1090.
[17] LI C H. Conditional stability of the general KdV soliton solution and KdV-Burgers traveling wave solution[J]. Acta Phys Sin, 1998, 47(9):1409-1415.
[18] QIN Y X, WANG M Q, WANG L. Motion stability theory and its application[M]. Beijing:Science Press, 1981:105-106.
[19] CALVO D C, YANG T S, AKYLAS T R. On the stability of solitary waves with decaying oscillatory tails[J]. Proc R Soc Lond A, 2000, 456:469-487.
[20] LU J F, GUAN Z. Numerical methods for solving partial differential equations[M]. Beijing:Tsinghua University Press, 2003:28-37.

Dynamical Stability of Traveling Wave Solutions to KdV-Burgers Equation

RichHTML

PDF

Knowledge

Abstract

Cite this article

share this article

References

Related Articles 2

Recommended Articles

Metrics

Authors

Reviewers

Journal Online

About Journal

[1]	PANG Jungang, SONG Lin, TANG Na, YANG Xueying, LI Xiaolin, XI Zhonghong, SHI Yuren. Shock Waves in Magnetized Dusty Plasma and Their Dynamical Stability Under Transverse Perturbations [J]. CHINESE JOURNAL OF COMPUTATIONAL PHYSICS, 2019, 36(6): 682-690.
[2]	WANG Linxue, ZONG Jin, WANG Xueling, SHI Yuren. Solitary Waves with Double Kinks of mBBM Equation and Their Dynamical Stabilities [J]. CHINESE JOURNAL OF COMPUTATIONAL PHYSICS, 2016, 33(2): 212-220.