中国物理B ›› 2003, Vol. 12 ›› Issue (9): 940-945.doi: 10.1088/1009-1963/12/9/303

• GENERAL • 上一篇    下一篇

Generalized Riccati equation expansion method and its application to the Bogoyavlenskii's generalized breaking soliton equation

陈勇, 李彪, 张鸿庆   

  1. Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, China; State Key Laboratory of Mathematics & Mechanization, Chinese Academy of Sciences, Beijing 100080, China
  • 收稿日期:2003-01-08 修回日期:2003-03-31 出版日期:2005-03-16 发布日期:2005-03-16
  • 基金资助:
    Project supported by the National Natural Science Foundation of China (Grant No 10072013), and the State Key Development Programme for Basic Research of China (Grant No G1998030600).

Generalized Riccati equation expansion method and its application to the Bogoyavlenskii's generalized breaking soliton equation

Chen Yong (陈勇), Li Biao (李彪), Zhang Hong-Qing (张鸿庆)   

  1. Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, China; State Key Laboratory of Mathematics & Mechanization, Chinese Academy of Sciences, Beijing 100080, China
  • Received:2003-01-08 Revised:2003-03-31 Online:2005-03-16 Published:2005-03-16
  • Supported by:
    Project supported by the National Natural Science Foundation of China (Grant No 10072013), and the State Key Development Programme for Basic Research of China (Grant No G1998030600).

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摘要： Based on the computerized symbolic system Maple and a Riccati equation, a Riccati equation expansion method is presented by a general ansatz. Compared with most of the existing tanh methods, the extended tanh-function method, the modified extended tanh-function method and generalized hyperbolic-function method, the proposed method is more powerful. By use of the method, we not only can successfully recover the previously known formal solutions but also construct new and more general formal solutions for some nonlinear differential equations. Making use of the method, we study the Bogoyavlenskii's generalized breaking soliton equation and obtain rich new families of the exact solutions, including the non-travelling wave and coefficient functions' soliton-like solutions, singular soliton-like solutions, periodic form solutions.

Abstract: Based on the computerized symbolic system Maple and a Riccati equation, a Riccati equation expansion method is presented by a general ansatz. Compared with most of the existing tanh methods, the extended tanh-function method, the modified extended tanh-function method and generalized hyperbolic-function method, the proposed method is more powerful. By use of the method, we not only can successfully recover the previously known formal solutions but also construct new and more general formal solutions for some nonlinear differential equations. Making use of the method, we study the Bogoyavlenskii's generalized breaking soliton equation and obtain rich new families of the exact solutions, including the non-travelling wave and coefficient functions' soliton-like solutions, singular soliton-like solutions, periodic form solutions.

Key words: generalized Riccati equation expansion, Bogoyavlenskii's generalized breaking soliton equation, soliton-like solution, periodic form solution

中图分类号:  (Ordinary differential equations)

  • 02.30.Hq
02.30.Jr (Partial differential equations) 02.60.-x (Numerical approximation and analysis)