Nonlinear Schrödinger equations play essential roles in different physics and engineering fields. In this paper, a hyper-block finite-difference self-consistent method (HFDSCF) is employed to solve this stationary nonlinear eigenvalue equation and demonstrated its accuracy. By comparing the results with the Sinc self-consistent (SSCF) method and the exact available results, we show that the HFDSCF gives quantum states with high accuracy and can even solve the strongly nonlinear Schrodinger equations. Then, by applying our method to the Hofstadter butterfly problem, we describe the breeding, metamorphosis, and killing of these butterflies by using nonlinear interactions and two constant length multi-well and sinusoidal potentials.

非线性Schrödinger方程在不同的物理和工程领域中发挥着重要作用。本文采用超块有限差分自洽方法（HFDSCF）求解该平稳非线性特征值方程，并证明了其准确性。通过将结果与Sinc自洽（SSCF）方法进行比较以及确切的可用结果，我们表明HFDSCF可以给出高精度的量子态，甚至可以求解强非线性Schrodinger方程。然后，通过将我们的方法应用于霍夫施塔特蝴蝶问题，我们利用非线性相互作用以及两个恒定长度的多阱和正弦势来描述这些蝴蝶的繁殖，变态和杀死。