具5次强非线性项的广义对称正则长波方程的显式精确解

Precise and Explicit Solutions to a Generalized Symmetric Regularized Long Wave Equations with Five Order Stronger Nonlinear Term

考虑了一类具５ 次强非线性项的广义对称正则长波方程的精确可解性问题。首先将求此方程孤立波解的问题归结为求解具５ 次强非线性项的Ｌｉｅｎａｒｄ 方程ｕ″（ξ） ＋ ｌｕ（ ξ） ＋ ｍ ｕ３（ξ） ＋ ｎｕ５（ξ） ＝０ ，接着通过变换ｕ（ξ） ＝ φ（ξ） 得到φ（ ξ） 满足的方程２ φ（ξ） φ″（ ξ） － φ′２（ξ） ＋ ４ｌφ２（ ξ） ＋ ４ ｍφ３（ξ） ＋４ ｎφ４（ξ） ＝ ０ ，最后通过两种假设φ（ξ） ＝ Ａｅα（ξ＋ ξ０）／［（１ ＋ ｅα（ξ＋ ξ０））２ ＋ Ｂｅα（ξ＋ ξ０）］ 和φ（ξ） ＝ Ａｅα（ ξ＋ ξ０）／（１ ＋ｅα（ξ＋ ξ０）） 获得了５ 次Ｌｉｅｎａｒｄ 方程的二类显式精确解。据此求出了具５ 次非线性项的广义对称正则长波方程的钟状和扭状孤立波解。

Precise soluability to a class of generalized symmetric regularized long wave equations with five order stronger nonlinear term are considered. Firstly, the problem of seek to solitary wave solutions of the equation are reduced to a problem to solve lienard equation with five order stronger nonlinear term u″(ξ)+lu(ξ)+mu 3(ξ)+nu 5(ξ) =0. Then, a equation 2 φ(ξ)φ″(ξ)-φ′ 2(ξ)+4lφ 2(ξ)+4mφ 3(ξ)+4nφ 4(ξ) =0 is obtained by a transformation u(ξ)=φ(ξ) . Finally, precise and explicit solutions of two types for five order lienard equation are obtained by two different assumptions φ(ξ)=A e α(ξ+ξ 0) /[(1+ e α(ξ+ξ 0) ) 2+B e α(ξ+ξ 0) ] and φ(ξ)=A e α(ξ+ξ 0) /(1+ e α(ξ+ξ 0) ). According to this result, bell-shaped and kink-shaped solitary wave solutions for generalized symmetric regularized long wave equation are obtained, furthermore, singular travelling wave solutions of two types and periodic wave solutions of triangle function type are also given.