Dark solitons are common topological excitations in a wide array of nonlinear waves. The dark soliton excitation energy, crucial for exploring dark soliton dynamics, is necessarily calculated in a renormalized form due to its existence on a finite background. Despite its tremendous importance and success, the renormalized energy form was firstly only suggested with no detailed derivation, and was then "derived" in the grand canonical ensemble. In this work, we revisit this fundamental problem and provide an alternative and intuitive derivation of the energy form from the fundamental field energy by utilizing a limiting procedure that conserves number of particles. Our derivation yields the same result, putting therefore the dark soliton energy form on a solid basis.

中文翻译：

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暗孤子激发能量的直接推导

暗孤子是各种非线性波中常见的拓扑激发。暗孤子激发能量对于探索暗孤子动力学至关重要，由于其存在于有限背景中，因此必须以重整化形式计算。尽管它具有巨大的重要性和成功，但重整化的能量形式首先只是在没有详细推导的情况下被提出，然后在大正则集合中“推导”出来。在这项工作中，我们重新审视了这个基本问题，并通过利用粒子数守恒的限制程序从基本场能中提供了能量形式的替代和直观推导。我们的推导产生了相同的结果，因此将暗孤子能量形式建立在坚实的基础上。