Norm-Stabilized Imaginary-Time Evolution via Feedback Control

TL;DR

This paper introduces a norm-stabilized imaginary-time evolution method for the 1D nonlinear Schrödinger equation that uses adaptive feedback to maintain wavefunction norm without explicit renormalization, improving stability and flexibility.

Contribution

The authors develop a feedback-controlled ITE scheme that automatically stabilizes the wavefunction norm, eliminating the need for external normalization and enhancing convergence towards soliton solutions.

Findings

Effective stabilization of wavefunction norm without external renormalization.

Comparable accuracy to analytical solutions and baseline methods.

Framework adaptable to higher-dimensional systems.

Abstract

We present a norm-stabilized imaginary-time evolution (ITE) scheme for the one-dimensional nonlinear Schrodinger equation (NLSE). Traditional ITE solvers often require explicit renormalization of the wavefunction after each step to preserve norm, which can be disruptive and algorithmically inflexible. We propose an alternative approach in which the evolution is continuously stabilized using an adaptive feedback term mu(tau), proportional to the time derivative of the wavefunction norm. This results in a self-regulating flow that requires no external normalization while preserving convergence toward soliton solutions. We demonstrate the method's effectiveness by comparing the final wavefunction profiles and L2 errors against analytical solutions and baseline methods without feedback. Although this work focuses on the 1D case, the framework is designed to extend naturally to higher dimensions. Future work will explore the behavior of the feedback mechanism in 2D and 3D systems, multi-soliton scenarios, and external potentials.