标题： 通过反馈控制的归一化虚时演化 显示英文标题

标题： Norm-Stabilized Imaginary-Time Evolution via Feedback Control

摘要： 我们提出了一种针对一维非线性薛定谔方程（NLSE）的范数稳定虚时演化（ITE）方案。 传统的ITE求解器通常需要在每一步之后对波函数进行显式归一化以保持范数，这可能会产生干扰并导致算法不够灵活。 我们提出了一种替代方法，其中使用与波函数范数的时间导数成比例的自适应反馈项mu(tau)来持续稳定演化。 这导致了一种自我调节的流动，无需外部归一化即可保持收敛到孤子解。 我们通过将最终的波函数轮廓和L2误差与解析解和无反馈的基准方法进行比较，展示了该方法的有效性。 尽管本工作专注于一维情况，但该框架设计为可以自然扩展到更高维度。 未来的工作将探讨反馈机制在二维和三维系统、多孤子情形以及外势场中的行为。 显示英文摘要

摘要： 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.