The nonlinear Schrödinger equation with t-periodic data: II. Perturbative results
The nonlinear Schrödinger equation with t-periodic data: II. Perturbative results
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DOI：
10.1098/rspa.2014.0926
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发表时间：
2014-11
期刊：
影响因子：
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通讯作者：
J. Lenells;A. Fokas;A. Fokas
中科院分区：
文献类型：
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作者：
J. Lenells;A. Fokas;A. Fokas
本文考虑半直线上的非线性薛定谔方程，其Dirichlet边界条件对大的t趋于周期函数.我们假设这个函数足够小，也就是说，它可以表示为αg0b（t）的形式，其中α是一个小常数。假设Neumann边值在大的t时趋于周期函数g1b（t），我们证明了g1b（t）可以用α中的一个扰动级数来表示，该扰动级数可以显式地构造成任意阶。作为一个例子，我们计算g1b（t），对于g0b（t）是两个指数之和的特殊情况，其阶数为α8。我们还证明了存在特定的函数g0b（t），对于这些函数，上述级数可以求和，因此，对于这些函数，g1b（t）可以以封闭形式获得。最简单的此类函数是exp（iωt），其中ω是真实的常数。
We consider the nonlinear Schrödinger equation on the half-line with a given Dirichlet boundary datum which for large t tends to a periodic function. We assume that this function is sufficiently small, namely that it can be expressed in the form αg0b(t), where α is a small constant. Assuming that the Neumann boundary value tends for large t to the periodic function g1b(t), we show that g1b(t) can be expressed in terms of a perturbation series in α which can be constructed explicitly to any desired order. As an illustration, we compute g1b(t) to order α8 for the particular case that g0b(t) is the sum of two exponentials. We also show that there exist particular functions g0b(t) for which the above series can be summed up, and therefore, for these functions, g1b(t) can be obtained in closed form. The simplest such function is exp(iωt), where ω is a real constant.