一类不稳定的非线性Schrdinger方程的有限差分方法

Finite difference method for a class of nonlinear Schro ¨dinger equation

研究了一类不稳定的非线性Ｓｃｈｒｏ¨ｄｉｎｇｅｒ方程ｉｕｘ＋ｕｔｔ＋εｕｘｔ＋ｆ（｜ｕ｜２）ｕ＝０（ε１，ｘ∈［０，１］，０≤ｔ≤Ｔ）的初边值问题，构造了该问题的一类无条件稳定的差分格式，利用非线性函数的有界延拓法与能量估计法得到了差分格式的误差估计，证明了差分格式的收敛性与稳定性．

This paper considers the initial boundary value problem of a class of unstable nonlinear Schro ¨dinger equations i u x+u tt +εu xt +f(|u| 2)u=0(ε1,x∈,0≤t≤T) .For this purposes,a class of the unconditionally stable difference scheme is established.By using the bound extension method of nonlinear function and energy estimate method,the error estimate of the finite difference scheme is presented,the convergence and stability of the finite difference scheme are proved.