中国物理B ›› 2009, Vol. 18 ›› Issue (5): 1821-1827.doi: 10.1088/1674-1056/18/5/017

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Infinite series symmetry reduction solutions to the modified KdV--Burgers equation

焦小玉1, 姚若侠2, 楼森岳3   

  1. (1)Department of Physics, Shanghai Jiao Tong University, Shanghai 200062, China; (2)Department of Physics, Shanghai Jiao Tong University, Shanghai 200062, China *; (3)Department of Physics, Shanghai Jiao Tong University, Shanghai 200062, China;Department of Physics, Ningbo University, Ningbo 315211, China
  • 收稿日期:2008-07-03 修回日期:2008-08-05 出版日期:2009-05-20 发布日期:2009-05-20
  • 基金资助:
    Project supported by the National Natural Science Foundations of China (Grant Nos 10735030, 10475055, and 90503006), the National Basic Research Program of China (Grant No 2007CB814800), the Science Foundation for Post Doctorate Research from the Ministry

Infinite series symmetry reduction solutions to the modified KdV--Burgers equation

Yao Ruo-Xia(姚若侠)a)b)c)†, Jiao Xiao-Yu(焦小玉)a), and Lou Sen-Yue(楼森岳)a)c)   

  1. a Department of Physics, Shanghai Jiao Tong University, Shanghai 200062, China; b School of Computer Science, Shaanxi Normal University, Xi’an 710062, China; c Department of Physics, Ningbo University, Ningbo 315211, China
  • Received:2008-07-03 Revised:2008-08-05 Online:2009-05-20 Published:2009-05-20
  • Supported by:
    Project supported by the National Natural Science Foundations of China (Grant Nos 10735030, 10475055, and 90503006), the National Basic Research Program of China (Grant No 2007CB814800), the Science Foundation for Post Doctorate Research from the Ministry

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摘要： From the point of view of approximate symmetry, the modified Korteweg--de Vries--Burgers (mKdV--Burgers) equation with weak dissipation is investigated. The symmetry of a system of the corresponding partial differential equations which approximate the perturbed mKdV--Burgers equation is constructed and the corresponding general approximate symmetry reduction is derived; thereby infinite series solutions and general formulae can be obtained. The obtained result shows that the zero-order similarity solution to the mKdV--Burgers equation satisfies the Painlevé II equation. Also, at the level of travelling wave reduction, the general solution formulae are given for any travelling wave solution of an unperturbed mKdV equation. As an illustrative example, when the zero-order tanh profile solution is chosen as an initial approximate solution, physically approximate similarity solutions are obtained recursively under the appropriate choice of parameters occurring during computation.

关键词: modified Korteweg--de Vries--Burgers (mKdV--Burgers) equation, approximate symmetry reduction, series reduction solution

Abstract: From the point of view of approximate symmetry, the modified Korteweg--de Vries--Burgers (mKdV--Burgers) equation with weak dissipation is investigated. The symmetry of a system of the corresponding partial differential equations which approximate the perturbed mKdV--Burgers equation is constructed and the corresponding general approximate symmetry reduction is derived; thereby infinite series solutions and general formulae can be obtained. The obtained result shows that the zero-order similarity solution to the mKdV--Burgers equation satisfies the Painlevé II equation. Also, at the level of travelling wave reduction, the general solution formulae are given for any travelling wave solution of an unperturbed mKdV equation. As an illustrative example, when the zero-order tanh profile solution is chosen as an initial approximate solution, physically approximate similarity solutions are obtained recursively under the appropriate choice of parameters occurring during computation.

Key words: modified Korteweg--de Vries--Burgers (mKdV--Burgers) equation, approximate symmetry reduction, series reduction solution

中图分类号:  (Solitons)

  • 05.45.Yv
02.30.Jr (Partial differential equations)