The discretization of Gross-Pitaevskii equations (GPE) leads to a nonlinear eigenvalue problem with eigenvector nonlinearity (NEPv). In this paper, we use two Newton-based methods to compute the positive ground state of GPE. The first method comes from the Newton-Noda iteration for saturable nonlinear Schr\"odinger equations proposed by Liu, which can be transferred to GPE naturally. The second method combines the idea of the Bisection method and the idea of Newton method, in which, each subproblem involving block tridiagonal linear systems can be solved easily. We give an explicit convergence and computational complexity analysis for it. Numerical experiments are provided to support the theoretical results.
翻译:Gross-Pitaevskii 等式(GPE)的离散导致非线性二元值与电子元非线性(NEPv)的问题。 在本文中,我们使用基于牛顿的两种方法来计算GPE的正基状态。 第一种方法是刘家宝提议的可参数非线性非线性方程的牛顿-诺达迭代法,该方程可以自然地转移到GPE。 第二种方法结合了双切法和牛顿法的概念,其中涉及三对角线系统块形的每个子问题都可以很容易解决。 我们对此进行了明确的趋同和计算复杂性分析。 提供数字实验是为了支持理论结果。