KdV-Burgers方程行波解的稳定性
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摘要
对KdV-Burgers方程的行波解进行线性稳定性分析,数值结果表明:对于正耗散情形,其行波解是稳定的;对于负耗散情形,其行波解是不稳定的.其次构造有限差分法对其行波解进行非线性动力学演化,结果表明:对于正耗散情形,KdV-Burgers方程的行波解是稳定的.本文结果修正和完善了相关文献中所得结论.
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.
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.
引文
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[10]WANG L X,ZONG J,WANG X L,et al.Solitary waves with double kinks of m BBM 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 Kd V 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 Kd V type equation and traveling wave for the Kd V-Burgers type equation[J].Acta Phys Sin,1996,45(7):1087-1090.
[17]LI C H.Conditional stability of the general Kd V soliton solution and Kd V-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.
[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 Kd V-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.Kd V-Burgers equation modeling of turbulence[J].Science in China(A),1992,35(5):66-76.
[6]LIU S D,LIU S K.Kd V-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-Kd V type equation[J].Science in China(A),1987,1:64-73.
[8]LK P,SHI Y R,DUAN W S,et al.The solitary wave solutions to Kd V-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 m BBM 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 Kd V 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 Kd V type equation and traveling wave for the Kd V-Burgers type equation[J].Acta Phys Sin,1996,45(7):1087-1090.
[17]LI C H.Conditional stability of the general Kd V soliton solution and Kd V-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.