中国物理B ›› 2007, Vol. 16 ›› Issue (7): 1832-1837.doi: 10.1088/1009-1963/16/7/004

• GENERAL • 上一篇    下一篇

The classification of travelling wave solutions and superposition of multi-solutions to Camassa-- Holm equation with dispersion

刘成仕   

  1. Department of Mathematics, Daqing Petroleum Institute, Daqing 163318, China
  • 收稿日期:2006-09-25 修回日期:2006-11-06 出版日期:2007-07-04 发布日期:2007-07-04

The classification of travelling wave solutions and superposition of multi-solutions to Camassa-- Holm equation with dispersion

Liu Cheng-Shi(刘成仕)   

  1. Department of Mathematics, Daqing Petroleum Institute, Daqing 163318, China
  • Received:2006-09-25 Revised:2006-11-06 Online:2007-07-04 Published:2007-07-04

摘要: Under the travelling wave transformation, the Camassa--Holm equation with dispersion is reduced to an integrable ordinary differential equation (ODE), whose general solution can be obtained using the trick of one-parameter group. Furthermore, by using a complete discrimination system for polynomial, the classification of all single travelling wave solutions to the Camassa--Holm equation with dispersion is obtained. In particular, an affine subspace structure in the set of the solutions of the reduced ODE is obtained. More generally, an implicit linear structure in the Camassa--Holm equation with dispersion is found. According to the linear structure, we obtain the superposition of multi-solutions to Camassa--Holm equation with dispersion.

Abstract: Under the travelling wave transformation, the Camassa--Holm equation with dispersion is reduced to an integrable ordinary differential equation (ODE), whose general solution can be obtained using the trick of one-parameter group. Furthermore, by using a complete discrimination system for polynomial, the classification of all single travelling wave solutions to the Camassa--Holm equation with dispersion is obtained. In particular, an affine subspace structure in the set of the solutions of the reduced ODE is obtained. More generally, an implicit linear structure in the Camassa--Holm equation with dispersion is found. According to the linear structure, we obtain the superposition of multi-solutions to Camassa--Holm equation with dispersion.

Key words: classification of travelling wave solution, symmetry group, Camassa--Holm equation with dispersion, superposition of solutions

中图分类号:  (Ordinary differential equations)

  • 02.30.Hq
47.35.-i (Hydrodynamic waves)