A hyper-block self-consistent approach to nonlinear Schrodinger equations: Breeding, metamorphosis, and killing of Hofstadter Butterflies

Nonlinear Schrodinger equations play essential roles in different physics and engineering fields. In this paper, a hyper-block finite-difference self-consistent method (HFDSCF) is em-ployed to solve this stationary nonlinear eigenvalue equation and demonstrated its accu-racy. By comparing the results with the Sinc self-consistent (SSCF) method and the ex-act 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. (c) 2021 Elsevier B.V. All rights reserved.

Automated summary

The paper introduces a self-consistent method for solving nonlinear Schrodinger equations, demonstrating its high accuracy in providing quantum states and solving strong nonlinear problems. Additionally, the method is applied to the Hofstadter butterfly problem, illustrating the processes of breeding, metamorphosis, and killing using nonlinear interactions.