Efficient determination of the energy landscape of nonlinear Schr\"odinger-type equations

TL;DR

This paper presents an efficient numerical method for exploring the full energy landscape of nonlinear Schrödinger-type equations, including stable and unstable solutions, demonstrated on a complex 3D superconducting problem.

Contribution

The paper introduces a systematic approach using numerical parameter continuation to fully characterize solution spectra of nonlinear Schrödinger equations, including unstable states.

Findings

Successfully computed the complete spectrum of solutions in a 3D superconducting domain.

Revealed the complex landscape of stable and unstable solutions.

Demonstrated the method's efficiency for numerically demanding problems.

Abstract

We describe a systematic approach for the efficient numerical solution of nonlinear Schr\"odinger-type partial differential equations of the form $(K +V + g|\psi|^2)\psi=0$, with an energy operator $K$, a scalar potential $V$, and a scalar parameter $g$. Instrumental to the approach are developments in numerical linear and nonlinear algebra, specifically numerical parameter continuation. We demonstrate how a continuous sequence of solutions can be obtained regardless of their stability, so that finally the spectrum of stable and unstable solutions in the specified parameter range is fully revealed. The method is demonstrated for the GL equation in a three-dimensional superconducting domain with an inhomogeneous magnetic field, a numerically demanding problem known to have an involved solution landscape.