一类非线性常微分方程振荡解的渐近表示

The Asymptotic Representation of a Kind of the Nonlinear Ordinary Differential Equations

在本文中,我们讨论了非线性常微分方程y"=a0|x|αy3+a1|x|βy2+α2|x|γy+α3|x|δ振荡解的渐近表示．在这个方程中将α0,α,α1,β,α2,γ,α3,δ分别换成0,0,6,0,0,0,sgn(x),1就是著名的第一类Painleve方程,而将α0,α,α1,β,α2,γ,α3,δ分别换成2,0,0,0,sgn(x),1,α0,就是著名的第二类Painleve方程．当α0,α,α1,β,α2,γ,α3,δ分别换成-β/3γ,0,0,0,1/γ,1,α,0时,可用于组合KdV方程孤立子解的化简．

In this paper, some oscillating asymptotic solutions are discussed in a kind of the nonlinear ordinary differential equations. The equation is the first Painleve equation when α0,α,α1,β,α2,γ,α3,δ are replaced with 0, 0, 6, 0, 0, 0, sgn（x）, 1, respectively~ And it is the second Painleve equation when α0,α,α1,β,α2,γ,α3,δ are replaced with 2, 0, 0, 0, sgn（x）, 1, α, 0, respectively If α0,α,α1,β,α2,γ,α3,δ are replaced with - β/3γ, 0, 0, 0, 1/γ, 1, α, 0, respctively, the equation can be used to obtain the soditon solution for the combination KdV equation.