一类非线性初值问题的重正规化解

The Renormalization Solution for a Class of Nonlinear Initial Value Problem

利用重正规化方法,讨论了一类非线性初值问题.先用直接展开法求得方程:y″+py-εky3=f(x,ε),y(0)=A,y′(0)=B的带有长期项的解的渐近展开式,再用重正规化方法将所求解一致化,并将结果应用于文献[9]所讨论的问题,得到了文献[9]中问题的其它形式的解.它们具有两种不同的性态,但在初值为x(0)=0,x′(0)=0时,它们又有共同的周期,从而丰富了文献[9]中的相应结论.

This paper, using the renormalization method, discusses a class of nonlinear initial value problems. First, we obtain the asymptotic expansions of solution y″＋py-εky3=f（x,ε）,y（0）=A,y′= B for the problem using the direct expansion method, and then use the renormalization method to confirm the solution for the problem, and finally apply the results of this article for the problem that paper [9] has discussed, obtained the other forms of solution for the problem of paper[9], they have two dif- ferent behaviors, but their solutions have common cycle when the initial value is x（0）=0,S（0）=0, the correspondent results in paper [9] are popularized.