中国物理B ›› 2012, Vol. 21 ›› Issue (11): 110205-110205.doi: 10.1088/1674-1056/21/11/110205

• GENERAL • 上一篇    下一篇

Linear superposition solutions to nonlinear wave equations

刘煜   

  1. Henan Electric Power Research Institute, Zhengzhou 450052, China
  • 收稿日期:2012-04-12 修回日期:2012-05-22 出版日期:2012-10-01 发布日期:2012-10-01

Linear superposition solutions to nonlinear wave equations

Liu Yu (刘煜 )   

  1. Henan Electric Power Research Institute, Zhengzhou 450052, China
  • Received:2012-04-12 Revised:2012-05-22 Online:2012-10-01 Published:2012-10-01
  • Contact: Liu Yu E-mail:ly_hndl@yahoo.com.cn

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摘要： The solutions to a linear wave equation can satisfy the principle of superposition, i.e., the linear superposition of two or more known solutions is still a solution of the linear wave equation. We show in this article that many nonlinear wave equations possess exact traveling wave solutions involving hyperbolic, triangle, and exponential functions, and the suitable linear combinations of these known solutions can also constitute linear superposition solutions to some nonlinear wave equations with special structure characteristics. The linear superposition solutions to the generalized KdV equation K(2,2,1), the Oliver water wave equation, and the k(n,n) equation are given. The structure characteristic of the nonlinear wave equations having linear superposition solutions is analyzed, and the reason why the solutions with the forms of hyperbolic, triangle, and exponential functions can form the linear superposition solutions is also discussed.

关键词: linear superposition solution, nonlinear wave equation, generalized KdV equation, Oliver water wave equation

Abstract: The solutions to a linear wave equation can satisfy the principle of superposition, i.e., the linear superposition of two or more known solutions is still a solution of the linear wave equation. We show in this article that many nonlinear wave equations possess exact traveling wave solutions involving hyperbolic, triangle, and exponential functions, and the suitable linear combinations of these known solutions can also constitute linear superposition solutions to some nonlinear wave equations with special structure characteristics. The linear superposition solutions to the generalized KdV equation K(2,2,1), the Oliver water wave equation, and the k(n,n) equation are given. The structure characteristic of the nonlinear wave equations having linear superposition solutions is analyzed, and the reason why the solutions with the forms of hyperbolic, triangle, and exponential functions can form the linear superposition solutions is also discussed.

Key words: linear superposition solution, nonlinear wave equation, generalized KdV equation, Oliver water wave equation

中图分类号:  (Partial differential equations)

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