A Direct Derivation of the Dark Soliton Excitation Energy**Supported by the National Natural Science Foundation of China (Grant No. 11775176), the Basic Research Program of Natural Science of Shaanxi Province, China (Grant No. 2018KJXX-094), the Key Innovative Research Team of Quantum Many-Body Theory and Quantum Control in Shaanxi Province, China (Grant No. 2017KCT-12), and the Major Basic Research Program of

Abstract

Dark solitons are common topological excitations in a wide array of nonlinear waves. The dark soliton excitation energy is crucial for exploring dark soliton dynamics and 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 at first only suggested with no detailed derivation, and was then “derived” in the grand canonical ensemble. 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, thus putting the dark soliton energy form on a solid basis.