A Hyper-Block Self-Consistent Approach to Nonlinear Schrodinger
Equations: Breeding, Metamorphosis and Killing of Hofstadter Butterflies
Abstract
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 exact available results, we show that the HFDSCF method gives
quantum states with high accuracy and can even solve the strongly
nonlinear Schrodinger equations. Then, by applying our method to
Hofstadter butterfly problem, we describe the breeding, metamorphosis
and killing of these butterflies by using nonlinear interactions as well
as two constant length multi-well and sinusoidal potentials.