Newton conjugate gradient method for discrete nonlinear Schrödinger equations

Abstract

Discrete nonlinear Schrödinger equations (DNLSEs) are fundamental in describing wave dynamics in nonlinear lattices across various systems, including optics and cold atomic physics. With the advent of topological phases of matter, the DNLSEs that characterize nonlinear topological states and topological solitons in nonlinear topological systems have become increasingly complex. Newton’s method, which is a traditional approach to solve the DNLSEs, faces significant challenges in solving these intricate nonlinear problems. Here, we propose the Newton conjugate gradient (NCG) method as an efficient alternative for solving the DNLSEs. By combining Newton iterations with conjugate gradient iterations, the NCG method achieves a comparable number of iterations to Newton’s method but is significantly faster overall and better suited at finding complex solutions. Using bulk solitons in a nonlinear photonic Chern insulator as an example, we demonstrate that the NCG method is particularly well-suited for solving DNLSEs that describe nonlinear topological systems. The NCG method serves as a powerful tool for investigating nonlinear topological states and topological solitons in even more complex nonlinear topological systems.