Applied Mathematics and Mechanics

, Volume 21, Issue 12, pp 1371–1380 | Cite as

Numerical simulation of standing solitons and their interaction

  • Zhou Xian-chu
  • Rui Yi


Standing soliton was studied by numerical simulation of its governing equation, a cubic Schrödiger equation with a complex conjugate term, which was derived by Miles and was accepted. The value of linear damping in Miles equation was studied. Calculations showed that linear damping effects strongly on the formation of a standing soliton and Laedke and Spatschek stable condition is only a necessary condition, but not a sufficient one. The interaction of two standing solitons was simulated. Simulations showed that the interaction pattern depends on system parameters. Calculations for the different initial condition and its development indicated that a stable standing soliton can be formed only for proper initial disturbance, otherwise the disturbance will disappear or develop into several solitons.

Key words

soliton standing soliton cubic Schrödinger equation numerical simulation 

CLC number



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  1. [1]
    WU Jun-ru, Keolian R, Rudnick I. Observation of a non-propagating hydrodynamic soliton [J].Phys Rev Lett, 1984,52: 1421–1424.CrossRefGoogle Scholar
  2. [2]
    Larraza A, Putterman S. Theory of non-propagating surface-wave solitons [J].J Fluid Mech, 1984,148: 443–449.MATHCrossRefGoogle Scholar
  3. [3]
    Miles J W. Parametrically excited solitary waves [J].J Fluid Mech, 1984,148: 451–460.MATHMathSciNetCrossRefGoogle Scholar
  4. [4]
    Laedke E W, Spatschek K H. On localized solution in nonlinear Faraday resonance [J].J Fluid Mech, 1991,223: 589–601.MATHMathSciNetCrossRefGoogle Scholar
  5. [5]
    Chen X, Wei R J, Wang B R. Chaos in non-propagating hydrodynamics solitons [J].Phys Rev, 1996,53: 6016–6020.CrossRefGoogle Scholar
  6. [6]
    Chen W Z, Wei R J, Wang B R. Non-propagating interface solitary wave in fluid [J].Phys Lett A, 1995,208: 197–200.MATHMathSciNetCrossRefGoogle Scholar
  7. [7]
    Chen W Z. Experimental observation of self-localized structure in granular material [J].Phys Lett A, 1995,196: 321–325.CrossRefGoogle Scholar
  8. [8]
    CUI Hong-nong, et al. Observation and experiment of non-propagating solitary wave [J].Acta of Xiangtan University in Nature Science, 1986,4: 27–34. (in Chinese)Google Scholar
  9. [9]
    CUI Hong-nong, et al. Study on the characteristic of non-propagating solitary wave [J].J Hydrodynamics, 1991,6(1): 18–25. (in Chinese)Google Scholar
  10. [10]
    ZHOU Xian-chu, CUI Hong-nong. The effect of surface tension no non-propagating solitary waves [J].Science in China A, 1992,12: 1269–1276.Google Scholar
  11. [11]
    ZHOU Xian-chu. Non-propagating soliton and surface tension [J].Acta Mech Sinica, 1998,30 (6): 672–675. (in Chinese)Google Scholar
  12. [12]
    YAN Jia-ren, HUANG Guo-xiang. Non-propagating solitary wave on interface of two layer fluid in rectangular wave guide [J].Acta Phys, 1988,37: 874–880. (in Chinese)Google Scholar
  13. [13]
    Yan J R, Mei Y P. Interaction between two Wu's solitons [J].Europhys Lett, 1993,23: 335–340.Google Scholar
  14. [14]
    ZHOU Xian-chu, TANG Shi-min, QIN Su-di. The stability of a standing soliton [A]. In: CHIEN Wei-zang, GUO Zhong-heng, GUO You-zhong Eds.Proc 2 nd Int Conf of Nonlinear Mech[C]. Beijing: Peking University Press, 1993, 455–458.Google Scholar

Copyright information

© Editorial Committee of Applied Mathematics and Mechanics 2000

Authors and Affiliations

  • Zhou Xian-chu
    • 1
  • Rui Yi
    • 1
  1. 1.Institute of MechanicsChinese Academy of SciencesBeijingP R China

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